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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 4
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 4
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 4
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000824
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000824: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 86%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000824: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 86%
Values
[1] => [.,.]
=> [1,0]
=> [1] => ? = 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,2,1] => 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 4
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 4
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 6
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 4
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 6
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 4
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 6
[1,4,5,3,2,6,7] => [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => ? = 4
[1,4,5,3,2,7,6] => [.,[[[.,[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,5,3,4,2,7,6] => ? = 6
[1,4,5,6,3,2,7] => [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => ? = 4
[1,4,5,6,7,3,2] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 4
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 6
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 8
[1,5,6,3,4,2,7] => [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => ? = 4
[1,5,6,4,2,3,7] => [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => ? = 4
[1,5,6,4,3,2,7] => [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 6
[1,5,6,7,3,4,2] => [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,6,3,4,5,7,2] => ? = 4
[1,5,6,7,4,2,3] => [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => ? = 4
[1,5,6,7,4,3,2] => [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? = 6
[1,6,4,5,3,2,7] => [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => ? = 6
[1,6,5,3,4,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? = 6
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 6
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 8
[1,6,7,3,4,5,2] => [.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,5,3,4,6,7,2] => ? = 4
[1,6,7,4,5,2,3] => [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => ? = 4
[1,6,7,4,5,3,2] => [.,[[[[.,[.,.]],[.,.]],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,7,4,5,3,6,2] => ? = 6
[1,6,7,5,2,3,4] => [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => ? = 4
[1,6,7,5,3,4,2] => [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 6
[1,6,7,5,4,2,3] => [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 6
[1,6,7,5,4,3,2] => [.,[[[[[.,[.,.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? = 8
[1,7,4,5,6,3,2] => [.,[[[[.,.],[.,[.,.]]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => ? = 6
[1,7,5,6,3,4,2] => [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,6,4,3,5,7,2] => ? = 6
[1,7,5,6,4,2,3] => [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => ? = 6
[1,7,5,6,4,3,2] => [.,[[[[[.,.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => ? = 8
[1,7,6,3,4,5,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 6
[1,7,6,4,5,2,3] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? = 6
[1,7,6,4,5,3,2] => [.,[[[[[.,.],.],[.,.]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,7,5,4,3,6,2] => ? = 8
[1,7,6,5,2,3,4] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 6
[1,7,6,5,3,4,2] => [.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 8
[1,7,6,5,4,2,3] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 8
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 10
[2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 2
[2,1,3,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 4
[2,1,3,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 4
[2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 4
[2,1,3,4,7,5,6] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 4
[2,1,3,4,7,6,5] => [[.,.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => ? = 6
[2,1,3,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 4
[2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 6
[2,1,3,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 4
[2,1,3,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 4
[2,1,3,6,4,5,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 4
[2,1,3,6,5,4,7] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => ? = 6
[2,1,3,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 4
[2,1,3,6,7,5,4] => [[.,.],[.,[[[.,[.,.]],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,7,5,6,4] => ? = 6
[2,1,3,7,4,5,6] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 4
Description
The sum of the number of descents and the number of recoils of a permutation.
This statistic is the sum of [[St000021]] and [[St000354]].
Matching statistic: St000915
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 86%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
[1,2,3,4,5,6,7] => [7] => ([],7)
=> ?
=> ? = 0
[1,2,3,4,5,7,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,6,5,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,6,7,5] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,7,5,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,7,6,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,4,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,5,6,4,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,6,7,4] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,5,4,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,6,7,4,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,7,5,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,5,6,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,7,6,4,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,3,7,6,5,4] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,4,3,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,3,5,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,4,3,6,5,7] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,4,3,6,7,5] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,4,3,7,5,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,4,3,7,6,5] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,3,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,4,5,6,3,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,6,7,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,5,4,3,7,6] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,6,4,3,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,5,6,7,3,4] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,6,7,4,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,6,3,4,5,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,6,4,5,3,7] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,6,5,3,4,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,6,5,4,3,7] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,6,7,4,5,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,6,7,5,3,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,6,7,5,4,3] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,7,3,4,5,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,7,4,5,6,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,7,5,6,3,4] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,7,5,6,4,3] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
[1,2,7,6,3,4,5] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4
[1,2,7,6,4,5,3] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6
Description
The Ore degree of a graph.
This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.
Matching statistic: St001500
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,3,4,6,5,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,3,4,7,5,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,3,4,7,6,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,3,5,6,4,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,3,6,7,4,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,3,6,7,5,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,3,7,6,5,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4 + 1
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4 + 1
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 6 + 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,4,5,6,3,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4 + 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 6 + 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,5,6,4,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,5,6,7,3,4] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,5,6,7,4,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,5,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,6,5,3,4,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,2,6,5,4,3,7] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 6 + 1
[1,2,6,7,3,4,5] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,6,7,4,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,6,7,5,3,4] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,6,7,5,4,3] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,2,7,3,4,5,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,4,5,6,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4 + 1
[1,2,7,5,6,3,4] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,2,7,5,6,4,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 6 + 1
[1,2,7,6,3,4,5] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,2,7,6,4,5,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 6 + 1
Description
The global dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
Matching statistic: St001278
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 6
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,6,4,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,2,3,6,5,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,2,4,5,6,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4
[1,2,5,6,3,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,2,5,6,4,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,3,4,5,2,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,3,4,5,6,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 6
[1,4,5,2,3,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4
[1,4,5,6,2,3] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[1,4,5,6,3,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,5,2,3,4,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,5,3,4,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4
[1,5,4,2,3,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4
[1,5,4,3,2,6] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 6
[1,5,6,2,3,4] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,5,6,3,4,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,5,6,4,2,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4
[1,5,6,4,3,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 6
[1,6,2,3,4,5] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,6,3,4,5,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,6,4,5,2,3] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4
[1,6,4,5,3,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 6
[1,6,5,2,3,4] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4
[1,6,5,3,4,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 6
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St001893
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001893: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 57%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001893: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [3,2,1] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 4
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 4
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 4
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 4
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 4
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 6
[2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 4
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 6
[3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[3,4,2,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 4
[4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 2
[4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 4
[4,3,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 4
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 6
[4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[4,5,2,3,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 4
[4,5,3,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 4
[5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[5,2,3,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4
[5,3,4,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 4
[5,3,4,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 6
[5,4,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 4
[5,4,2,3,1] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 6
[5,4,3,1,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 6
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 8
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 4
[1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 4
[1,2,5,6,3,4] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 2
[1,2,5,6,4,3] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 4
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 2
[1,2,6,4,5,3] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 4
[1,2,6,5,3,4] => [1,2,6,5,3,4] => [1,2,6,5,3,4] => ? = 4
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 6
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
[1,3,2,5,6,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => ? = 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 6
Description
The flag descent of a signed permutation.
$$ fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 ) $$
It has the same distribution as the flag excedance statistic.
Matching statistic: St001892
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 57%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 4
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[4,2,3,1] => [4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 4
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 4
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 4
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,5,3,4,2] => [1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 4
[1,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 6
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4
[2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 4
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 6
[2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 4
[3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 6
[3,4,1,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[3,4,2,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 4
[4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 2
[4,2,3,1,5] => [4,1,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 4
[4,3,1,2,5] => [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 4
[4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[4,5,1,2,3] => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 2
[4,5,2,3,1] => [2,5,1,4,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 4
[4,5,3,1,2] => [2,5,4,1,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 4
[5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[5,2,3,4,1] => [5,1,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 4
[5,3,4,1,2] => [5,2,4,1,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4
[5,3,4,2,1] => [5,1,4,3,2] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 6
[5,4,1,2,3] => [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 4
[5,4,2,3,1] => [5,4,1,3,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 6
[5,4,3,1,2] => [5,4,3,1,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 6
[5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 4
[1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? = 4
[1,2,5,6,3,4] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => [1,2,6,4,3,5] => ? = 2
[1,2,5,6,4,3] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 4
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 2
[1,2,6,4,5,3] => [1,2,6,3,5,4] => [1,2,6,5,3,4] => [1,2,6,5,3,4] => ? = 4
[1,2,6,5,3,4] => [1,2,6,5,3,4] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 4
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? = 6
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
[1,3,2,5,6,4] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 4
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => ? = 4
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,5,6,4] => ? = 6
Description
The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
Matching statistic: St000524
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000524: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000524: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0 - 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0 - 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,1),(0,2),(1,12),(1,14),(2,11),(2,13),(3,145),(4,146),(5,31),(5,155),(6,32),(6,156),(7,15),(7,143),(8,16),(8,144),(9,37),(9,114),(10,38),(10,115),(11,17),(11,157),(12,18),(12,158),(13,29),(13,35),(13,157),(14,30),(14,36),(14,158),(15,84),(16,85),(17,147),(18,148),(19,102),(19,133),(20,103),(20,134),(21,23),(21,149),(21,153),(22,24),(22,150),(22,154),(23,118),(23,128),(24,119),(24,129),(25,62),(25,82),(26,63),(26,83),(27,74),(27,139),(28,75),(28,140),(29,100),(29,126),(30,101),(30,127),(31,55),(31,135),(32,56),(32,136),(33,47),(33,72),(33,98),(34,48),(34,73),(34,99),(35,100),(35,130),(35,132),(36,101),(36,131),(36,132),(37,96),(37,110),(37,112),(38,97),(38,111),(38,113),(40,201),(40,202),(41,225),(42,226),(43,221),(43,223),(44,222),(44,224),(45,213),(45,221),(46,214),(46,222),(47,213),(47,223),(48,214),(48,224),(49,161),(50,162),(51,183),(51,227),(52,184),(52,228),(53,173),(53,195),(54,174),(54,196),(55,185),(56,186),(57,179),(57,180),(58,175),(58,177),(59,176),(59,178),(60,171),(60,177),(61,172),(61,178),(62,215),(63,216),(64,220),(65,219),(66,219),(67,220),(68,183),(68,203),(69,184),(69,204),(70,179),(70,199),(71,180),(71,200),(72,9),(72,223),(73,10),(73,224),(74,3),(74,189),(75,4),(75,190),(76,53),(76,205),(76,227),(77,54),(77,206),(77,228),(78,53),(78,207),(79,54),(79,208),(80,58),(80,163),(80,167),(81,59),(81,164),(81,168),(82,49),(82,215),(83,50),(83,216),(84,94),(85,95),(86,68),(86,197),(87,69),(87,198),(88,94),(89,95),(90,55),(90,169),(91,56),(91,170),(92,108),(92,170),(93,109),(93,169),(94,39),(95,39),(96,86),(96,163),(97,87),(97,164),(98,19),(98,137),(98,213),(99,20),(99,138),(99,214),(100,21),(100,116),(100,159),(101,22),(101,117),(101,160),(102,90),(102,217),(103,91),(103,218),(104,51),(104,181),(105,52),(105,182),(106,41),(106,187),(107,42),(107,188),(108,40),(108,175),(108,194),(109,40),(109,176),(109,193),(110,60),(110,163),(110,191),(111,61),(111,164),(111,192),(112,104),(112,191),(113,105),(113,192),(114,112),(115,113),(116,124),(116,149),(117,124),(117,150),(118,91),(118,92),(119,90),(119,93),(120,58),(120,60),(120,225),(121,59),(121,61),(121,226),(122,92),(122,125),(122,218),(123,93),(123,125),(123,217),(124,122),(124,123),(125,108),(125,109),(125,167),(125,168),(126,43),(126,72),(126,159),(127,44),(127,73),(127,160),(128,41),(128,120),(129,42),(129,121),(130,45),(130,98),(130,159),(131,46),(131,99),(131,160),(132,116),(132,117),(133,80),(133,96),(133,217),(134,81),(134,97),(134,218),(135,51),(135,76),(135,185),(136,52),(136,77),(136,186),(137,133),(137,151),(137,165),(138,134),(138,152),(138,166),(139,57),(139,70),(139,189),(140,57),(140,71),(140,190),(141,76),(141,78),(141,181),(142,77),(142,79),(142,182),(143,84),(143,88),(144,85),(144,89),(145,64),(145,67),(146,65),(146,66),(147,43),(147,45),(147,47),(148,44),(148,46),(148,48),(149,103),(149,118),(149,122),(150,102),(150,119),(150,123),(151,80),(151,110),(151,120),(151,187),(152,81),(152,111),(152,121),(152,188),(153,106),(153,128),(153,151),(154,107),(154,129),(154,152),(155,104),(155,135),(155,141),(156,105),(156,136),(156,142),(157,33),(157,126),(157,130),(157,147),(158,34),(158,127),(158,131),(158,148),(159,137),(159,153),(159,221),(160,138),(160,154),(160,222),(161,143),(162,144),(163,27),(163,177),(163,197),(164,28),(164,178),(164,198),(165,155),(165,187),(166,156),(166,188),(167,175),(167,193),(167,197),(168,176),(168,194),(168,198),(169,68),(169,185),(169,193),(170,69),(170,186),(170,194),(171,207),(172,208),(173,215),(173,231),(174,216),(174,232),(175,201),(175,209),(176,202),(176,210),(177,74),(177,209),(178,75),(178,210),(179,66),(179,233),(180,67),(180,233),(181,25),(181,207),(181,227),(182,26),(182,208),(182,228),(183,229),(184,230),(185,183),(185,205),(186,184),(186,206),(187,141),(187,191),(187,225),(188,142),(188,192),(188,226),(189,145),(189,180),(189,199),(190,146),(190,179),(190,200),(191,171),(191,181),(192,172),(192,182),(193,201),(193,203),(193,205),(194,202),(194,204),(194,206),(195,231),(196,232),(197,139),(197,203),(197,209),(198,140),(198,204),(198,210),(199,64),(199,233),(200,65),(200,233),(201,195),(201,211),(202,196),(202,212),(203,70),(203,211),(203,229),(204,71),(204,212),(204,230),(205,195),(205,229),(206,196),(206,230),(207,62),(207,173),(208,63),(208,174),(209,189),(209,211),(210,190),(210,212),(211,199),(211,231),(212,200),(212,232),(213,5),(213,165),(214,6),(214,166),(215,7),(215,161),(216,8),(216,162),(217,86),(217,167),(217,169),(218,87),(218,168),(218,170),(219,88),(220,89),(221,106),(221,165),(222,107),(222,166),(223,114),(224,115),(225,78),(225,171),(226,79),(226,172),(227,82),(227,173),(227,229),(228,83),(228,174),(228,230),(229,49),(229,231),(230,50),(230,232),(231,161),(232,162),(233,219),(233,220)],234)
=> ? = 4 - 1
[4,5,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,18),(0,19),(1,17),(1,50),(2,10),(2,13),(3,63),(3,272),(4,284),(5,283),(6,64),(6,287),(7,8),(7,325),(8,309),(9,60),(9,314),(10,38),(10,310),(11,40),(11,149),(12,49),(12,59),(12,321),(13,22),(13,58),(13,310),(14,43),(14,278),(15,41),(15,280),(16,42),(16,313),(17,34),(17,35),(17,304),(18,1),(18,184),(19,12),(19,37),(19,184),(20,231),(20,235),(21,233),(21,298),(22,238),(22,294),(23,68),(23,318),(23,324),(24,123),(24,241),(25,226),(25,237),(26,52),(26,229),(26,322),(27,124),(27,230),(28,127),(28,302),(29,125),(29,303),(30,120),(30,296),(31,232),(31,240),(32,225),(32,293),(33,224),(33,292),(34,243),(34,295),(35,243),(35,316),(36,227),(36,289),(37,51),(37,290),(37,321),(38,242),(38,320),(39,219),(39,300),(40,222),(40,223),(41,220),(41,221),(42,121),(42,297),(43,122),(43,228),(44,66),(44,236),(44,291),(45,65),(45,126),(45,234),(46,67),(46,288),(46,299),(47,54),(47,301),(47,317),(48,55),(48,319),(48,323),(49,53),(49,239),(49,315),(50,242),(50,304),(51,167),(51,308),(52,206),(52,273),(53,255),(53,263),(54,157),(54,253),(55,254),(55,266),(56,181),(56,183),(56,218),(57,182),(57,267),(57,270),(58,238),(58,269),(58,271),(59,239),(59,268),(59,271),(60,106),(60,180),(60,274),(61,89),(61,215),(61,217),(62,90),(62,119),(62,216),(63,145),(63,189),(63,194),(64,57),(64,255),(64,306),(64,316),(65,155),(65,195),(65,197),(66,153),(66,154),(66,156),(67,101),(67,196),(67,262),(68,131),(68,244),(68,245),(69,452),(70,466),(71,417),(72,454),(72,456),(73,451),(73,453),(74,426),(74,449),(75,425),(75,455),(76,414),(76,416),(77,336),(77,456),(78,336),(78,454),(79,335),(79,344),(80,342),(80,450),(81,343),(81,453),(82,11),(83,415),(83,467),(84,413),(84,467),(85,455),(85,465),(86,341),(86,464),(87,472),(88,471),(89,342),(89,441),(90,3),(90,451),(91,357),(92,358),(93,358),(94,394),(94,406),(95,364),(95,452),(96,372),(96,468),(97,371),(97,457),(98,364),(98,424),(99,359),(99,404),(100,376),(100,400),(101,326),(101,377),(102,332),(102,363),(103,331),(103,351),(104,332),(104,374),(105,348),(105,463),(106,341),(106,375),(107,350),(108,409),(109,410),(110,459),(111,459),(112,421),(112,468),(113,458),(114,397),(115,356),(115,408),(116,371),(116,418),(117,372),(117,420),(118,370),(118,411),(119,36),(119,343),(119,451),(120,432),(121,393),(122,369),(123,391),(124,4),(124,392),(125,5),(125,390),(126,46),(126,368),(127,39),(127,365),(128,99),(129,150),(130,149),(131,264),(131,461),(132,178),(132,470),(133,212),(133,462),(134,218),(134,447),(134,449),(135,138),(136,198),(136,436),(137,91),(137,458),(138,92),(138,422),(139,99),(139,407),(139,465),(140,202),(140,464),(141,118),(141,469),(142,97),(142,435),(143,188),(144,100),(144,419),(144,457),(145,200),(145,433),(146,102),(146,345),(146,433),(147,103),(147,346),(147,434),(148,126),(148,344),(149,223),(150,92),(151,193),(152,122),(152,423),(153,163),(153,338),(154,209),(154,337),(155,201),(155,434),(156,240),(156,337),(156,338),(157,265),(158,221),(158,471),(159,93),(160,93),(161,181),(161,444),(161,447),(162,94),(162,405),(162,437),(163,180),(163,438),(164,130),(164,438),(165,175),(165,460),(166,176),(166,337),(166,460),(167,252),(168,106),(168,340),(169,121),(169,339),(170,165),(170,442),(171,210),(171,340),(172,211),(172,339),(173,98),(173,328),(174,79),(174,441),(174,450),(175,78),(175,329),(175,443),(176,77),(176,440),(176,443),(177,69),(177,439),(178,71),(178,328),(179,74),(179,335),(179,432),(180,199),(180,341),(181,288),(181,329),(182,131),(182,327),(182,442),(183,72),(183,329),(183,440),(184,6),(184,290),(185,231),(185,355),(186,203),(186,334),(186,385),(187,105),(187,388),(188,214),(188,382),(189,136),(189,381),(190,213),(190,385),(191,100),(191,389),(192,94),(192,378),(193,108),(193,357),(194,104),(194,381),(194,433),(195,142),(195,384),(196,188),(196,377),(197,205),(197,384),(197,434),(198,222),(198,356),(198,437),(199,186),(199,383),(200,115),(200,367),(201,116),(201,366),(202,109),(202,349),(203,95),(203,380),(203,439),(204,207),(204,382),(205,101),(205,331),(205,395),(206,71),(206,352),(207,83),(207,360),(207,403),(208,70),(208,387),(209,86),(209,362),(210,76),(210,351),(210,401),(211,76),(211,363),(211,402),(212,75),(212,326),(212,349),(213,98),(213,354),(213,380),(214,84),(214,359),(214,360),(215,33),(215,286),(215,342),(216,32),(216,285),(216,343),(217,45),(217,148),(217,441),(218,21),(218,247),(218,440),(219,151),(220,150),(221,159),(222,114),(222,334),(223,177),(223,334),(224,169),(224,446),(225,168),(225,445),(226,143),(227,82),(228,138),(228,369),(229,206),(229,396),(230,107),(230,392),(231,91),(231,373),(232,130),(232,353),(233,132),(233,361),(234,195),(234,368),(235,152),(235,373),(236,31),(236,156),(236,472),(237,26),(237,250),(238,48),(238,249),(238,330),(239,23),(239,248),(239,333),(240,47),(240,276),(240,353),(241,20),(241,185),(241,391),(242,261),(243,7),(243,307),(244,105),(244,256),(244,461),(245,168),(245,171),(246,102),(246,104),(246,466),(247,212),(247,298),(247,347),(248,259),(248,318),(249,259),(249,319),(250,204),(250,322),(251,173),(251,213),(251,470),(252,1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(469,370),(470,141),(470,328),(470,354),(471,159),(471,160),(472,164),(472,232),(472,338),(473,471),(473,481),(474,250),(474,462),(475,458),(475,459),(476,107),(476,478),(477,355),(478,350),(479,88),(479,473),(480,151),(480,469),(481,160),(482,185),(482,477)],483)
=> ? = 6 - 1
[5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,23),(0,24),(1,14),(2,13),(3,35),(3,78),(4,36),(4,79),(5,33),(5,54),(6,34),(6,55),(7,15),(7,80),(8,16),(8,81),(9,17),(9,52),(10,18),(10,53),(11,19),(11,31),(11,82),(12,20),(12,32),(12,83),(13,76),(14,77),(15,58),(16,59),(17,91),(18,92),(19,21),(19,74),(20,22),(20,75),(21,27),(21,88),(22,28),(22,89),(23,11),(23,90),(24,12),(24,90),(25,68),(25,101),(26,69),(26,102),(27,66),(27,97),(28,67),(28,98),(29,47),(29,99),(30,46),(30,100),(31,56),(31,74),(32,57),(32,75),(33,30),(33,97),(33,104),(34,29),(34,98),(34,104),(35,25),(35,84),(35,86),(36,26),(36,85),(36,87),(38,121),(38,122),(39,107),(39,123),(40,108),(40,124),(41,107),(41,108),(42,125),(43,126),(44,116),(44,121),(45,115),(45,122),(46,117),(47,118),(48,125),(49,126),(50,113),(51,114),(52,2),(53,1),(54,3),(55,4),(56,54),(57,55),(58,52),(59,53),(60,50),(60,119),(61,51),(61,120),(62,76),(62,119),(63,77),(63,120),(64,37),(65,37),(66,78),(67,79),(68,62),(68,105),(69,63),(69,106),(70,42),(70,111),(71,43),(71,112),(72,44),(72,110),(73,45),(73,109),(74,7),(74,88),(75,8),(75,89),(76,93),(77,94),(78,84),(79,85),(80,9),(80,58),(81,10),(81,59),(82,5),(82,56),(83,6),(83,57),(84,68),(84,91),(84,123),(85,69),(85,92),(85,124),(86,73),(86,101),(86,123),(87,72),(87,102),(87,124),(88,66),(88,80),(89,67),(89,81),(90,82),(90,83),(91,60),(91,62),(92,61),(92,63),(93,42),(93,48),(94,43),(94,49),(95,39),(95,41),(95,117),(96,40),(96,41),(96,118),(97,46),(97,95),(98,47),(98,96),(99,39),(99,86),(99,118),(100,40),(100,87),(100,117),(101,38),(101,45),(101,105),(102,38),(102,44),(102,106),(103,64),(103,65),(104,95),(104,96),(104,99),(104,100),(105,115),(105,119),(105,121),(106,116),(106,120),(106,122),(107,109),(108,110),(109,50),(109,115),(110,51),(110,116),(111,103),(111,125),(112,103),(112,126),(113,48),(113,111),(114,49),(114,112),(115,113),(115,127),(116,114),(116,127),(117,72),(117,108),(118,73),(118,107),(119,70),(119,93),(119,113),(120,71),(120,94),(120,114),(121,70),(121,127),(122,71),(122,127),(123,60),(123,105),(123,109),(124,61),(124,106),(124,110),(125,64),(126,65),(127,111),(127,112)],128)
=> ? = 4 - 1
[5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,25),(0,26),(0,66),(1,56),(2,55),(3,7),(3,200),(4,168),(5,303),(6,260),(7,264),(8,70),(8,306),(9,10),(9,68),(9,201),(10,18),(10,307),(11,48),(11,203),(12,47),(12,337),(13,52),(13,304),(14,53),(14,289),(15,46),(15,240),(16,44),(16,98),(17,54),(17,202),(18,45),(18,335),(19,51),(19,336),(20,24),(20,50),(20,204),(21,49),(21,344),(22,36),(22,338),(23,37),(23,38),(23,341),(24,34),(24,43),(24,253),(25,9),(25,276),(26,23),(26,40),(26,163),(27,256),(27,325),(28,130),(28,131),(29,76),(29,258),(29,327),(30,127),(30,323),(31,62),(31,129),(31,324),(32,75),(32,319),(32,328),(33,310),(33,326),(34,259),(34,342),(35,77),(35,245),(35,318),(36,242),(36,314),(37,133),(37,322),(38,64),(38,133),(38,339),(39,74),(39,254),(39,343),(40,73),(40,315),(40,341),(41,60),(41,257),(41,320),(42,59),(42,132),(42,329),(43,57),(43,259),(43,312),(44,63),(44,316),(44,345),(45,321),(45,330),(46,65),(46,330),(46,345),(47,251),(47,309),(48,246),(48,252),(49,241),(49,340),(50,249),(50,253),(51,250),(51,313),(52,244),(52,247),(53,126),(53,248),(54,61),(54,311),(54,317),(55,128),(55,255),(56,58),(56,243),(56,308),(57,162),(57,316),(58,216),(58,293),(59,214),(59,215),(60,159),(60,161),(61,160),(61,286),(62,140),(62,261),(63,155),(63,273),(64,274),(64,287),(65,96),(65,282),(66,20),(66,163),(66,276),(67,196),(67,198),(67,239),(68,197),(68,199),(68,307),(69,233),(69,234),(69,288),(70,72),(70,287),(70,334),(70,342),(71,125),(71,195),(71,238),(72,157),(72,267),(72,280),(73,173),(73,199),(73,333),(74,213),(74,236),(74,295),(75,102),(75,210),(75,262),(76,94),(76,150),(76,277),(77,69),(77,279),(77,291),(77,294),(78,432),(79,463),(80,465),(81,430),(82,366),(82,466),(83,367),(84,442),(84,464),(85,434),(85,481),(86,431),(87,480),(88,441),(89,438),(89,470),(90,429),(90,433),(91,364),(91,473),(92,354),(92,467),(93,354),(93,462),(94,353),(94,365),(95,469),(95,470),(96,468),(97,487),(98,17),(99,359),(99,385),(100,382),(101,387),(101,425),(102,362),(102,404),(103,394),(103,436),(104,410),(104,435),(105,384),(105,419),(106,376),(106,423),(107,380),(107,474),(108,364),(108,389),(109,394),(109,482),(110,384),(110,476),(111,356),(111,401),(112,389),(112,453),(113,14),(113,391),(114,375),(114,387),(115,351),(116,427),(116,428),(117,426),(118,476),(119,488),(120,403),(121,381),(122,391),(123,385),(123,402),(124,388),(125,33),(125,357),(125,366),(126,352),(127,6),(127,446),(128,400),(129,13),(129,373),(130,5),(130,392),(131,4),(131,392),(132,32),(132,409),(133,19),(133,457),(134,191),(134,352),(135,153),(135,479),(136,218),(136,480),(137,232),(138,239),(138,459),(138,464),(139,171),(139,460),(140,175),(140,485),(141,128),(142,123),(142,478),(143,121),(143,463),(144,114),(144,439),(144,481),(145,100),(145,465),(146,99),(146,486),(147,237),(148,222),(148,475),(149,158),(149,447),(150,180),(150,353),(151,104),(151,351),(151,365),(152,111),(152,357),(152,466),(153,105),(153,350),(154,209),(154,474),(155,241),(155,363),(156,243),(157,250),(157,347),(157,460),(158,126),(158,349),(159,166),(159,360),(160,176),(160,346),(161,129),(161,360),(162,202),(163,8),(163,315),(164,255),(164,487),(165,106),(165,452),(166,242),(166,449),(167,318),(168,141),(169,101),(169,418),(170,134),(171,189),(171,472),(172,145),(172,450),(173,272),(173,451),(174,270),(174,449),(175,156),(176,251),(176,458),(177,120),(177,458),(178,99),(178,443),(178,483),(179,106),(179,437),(179,482),(180,308),(180,452),(181,142),(181,448),(182,190),(182,358),(182,472),(183,94),(183,462),(183,467),(184,86),(185,86),(185,440),(186,88),(186,349),(187,78),(188,79),(188,355),(189,93),(189,348),(189,455),(190,92),(190,454),(190,455),(191,80),(191,450),(192,84),(192,356),(192,446),(193,81),(193,350),(194,83),(194,456),(195,217),(195,366),(196,319),(196,348),(197,240),(197,451),(198,183),(198,348),(198,454),(199,266),(199,451),(200,12),(200,297),(201,15),(201,197),(202,317),(203,3),(203,246),(204,16),(204,249),(205,228),(205,396),(206,107),(206,408),(207,196),(207,390),(207,459),(208,110),(208,397),(209,102),(209,415),(210,221),(210,404),(211,104),(211,393),(211,469),(212,116),(212,395),(213,205),(213,414),(214,149),(214,406),(215,209),(215,406),(216,100),(216,412),(217,290),(217,407),(218,117),(218,371),(219,220),(219,379),(220,213),(220,410),(220,483),(221,235),(221,398),(222,103),(222,362),(222,371),(223,227),(223,398),(224,132),(224,401),(225,244),(225,378),(225,448),(226,87),(226,369),(227,85),(227,377),(227,424),(228,90),(228,375),(228,420),(229,88),(229,386),(230,89),(230,374),(231,78),(231,370),(232,81),(232,399),(233,225),(233,361),(233,413),(234,219),(234,413),(235,144),(235,376),(235,377),(236,105),(236,383),(236,414),(237,90),(237,381),(237,382),(238,42),(238,224),(238,357),(239,27),(239,268),(239,454),(240,35),(240,167),(241,300),(242,170),(243,216),(243,359),(244,284),(244,477),(245,294),(245,367),(246,200),(246,358),(247,285),(247,477),(248,172),(248,352),(249,98),(250,283),(250,471),(251,265),(252,226),(252,358),(253,11),(253,312),(254,236),(254,416),(255,231),(255,400),(256,135),(256,372),(257,31),(257,161),(257,456),(258,39),(258,269),(259,21),(259,332),(260,119),(260,278),(261,181),(261,225),(261,485),(262,109),(262,179),(262,362),(263,193),(263,232),(263,479),(264,176),(264,177),(265,110),(265,118),(266,194),(266,320),(267,182),(267,252),(267,460),(268,222),(268,325),(268,368),(269,223),(269,343),(270,175),(270,289),(271,180),(271,221),(272,127),(272,192),(273,112),(273,279),(273,363),(274,111),(274,224),(274,461),(275,108),(275,112),(275,468),(276,201),(276,204),(277,178),(277,220),(277,365),(278,219),(278,277),(278,488),(279,288),(279,298),(279,453),(280,84),(280,138),(280,347),(281,91),(281,108),(281,363),(282,91),(282,311),(282,468),(283,87),(283,136),(284,79),(284,143),(285,80),(285,145),(286,89),(286,95),(286,346),(287,139),(287,157),(287,461),(288,95),(288,211),(288,361),(289,156),(289,248),(290,154),(290,215),(290,408),(291,234),(291,278),(291,405),(292,228),(292,237),(292,411),(293,101),(293,114),(293,412),(294,233),(294,261),(294,405),(294,453),(295,85),(295,144),(295,383),(296,165),(296,179),(297,302),(297,337),(298,151),(298,211),(298,484),(299,186),(299,229),(299,447),(300,115),(300,151),(301,134),(301,331),(302,137),(302,263),(303,97),(303,164),(304,247),(304,331),(305,184),(305,185),(306,30),(306,272),(306,334),(307,41),(307,266),(307,335),(308,169),(308,293),(308,359),(309,153),(309,193),(310,166),(310,174),(311,230),(311,286),(311,473),(312,162),(312,203),(313,107),(313,154),(313,471),(314,158),(314,186),(315,173),(315,306),(316,155),(316,281),(317,160),(317,264),(317,473),(318,260),(318,291),(318,367),(319,262),(319,296),(320,159),(320,310),(320,456),(321,83),(321,245),(322,82),(322,195),(322,457),(323,67),(323,138),(323,207),(323,446),(324,140),(324,270),(324,373),(325,103),(325,109),(325,372),(326,113),(326,122),(327,150),(327,271),(328,210),(328,271),(329,113),(329,214),(329,409),(330,96),(330,275),(331,172),(331,191),(331,285),(332,171),(332,182),(332,344),(333,82),(333,125),(333,152),(334,192),(334,280),(334,323),(334,461),(335,194),(335,257),(335,321),(336,206),(336,290),(336,313),(337,135),(337,263),(337,309),(338,149),(338,299),(338,314),(339,152),(339,238),(339,274),(339,457),(340,92),(340,93),(340,183),(341,71),(341,322),(341,333),(341,339),(342,139),(342,267),(342,332),(343,227),(343,235),(343,295),(343,416),(344,189),(344,190),(344,198),(344,340),(345,273),(345,275),(345,281),(345,282),(346,438),(346,458),(346,469),(347,442),(347,459),(347,471),(348,296),(348,462),(349,131),(349,441),(350,116),(350,419),(350,430),(351,123),(351,435),(351,443),(352,450),(353,452),(354,117),(354,422),(355,305),(355,463),(356,390),(356,442),(357,326),(357,329),(357,401),(358,297),(358,369),(359,412),(359,418),(360,373),(360,449),(361,393),(361,448),(361,470),(362,394),(362,437),(363,298),(363,300),(363,389),(364,374),(365,410),(365,443),(366,22),(366,407),(367,119),(367,405),(368,371),(368,372),(368,422),(369,302),(369,480),(370,432),(371,426),(371,436),(371,437),(372,436),(372,479),(372,482),(373,301),(373,304),(374,120),(374,438),(375,429),(375,496),(376,439),(376,495),(377,303),(377,481),(377,495),(378,477),(378,486),(379,483),(379,486),(380,475),(381,433),(381,444),(382,429),(382,444),(383,419),(383,434),(383,439),(384,417),(385,418),(386,130),(386,441),(387,124),(387,496),(388,421),(389,115),(389,484),(390,327),(390,491),(391,1),(392,2),(393,435),(393,478),(393,494),(394,493),(395,427),(396,417),(396,420),(397,305),(397,476),(398,377),(399,395),(399,430),(400,370),(401,122),(401,409),(402,121),(402,411),(403,118),(403,397),(404,398),(405,413),(405,485),(405,488),(406,415),(406,447),(407,338),(407,408),(408,299),(408,406),(408,474),(409,328),(409,391),(410,414),(410,492),(411,381),(411,420),(412,375),(412,382),(412,425),(413,378),(413,379),(414,384),(414,396),(415,386),(415,404),(416,376),(416,383),(416,424),(417,440),(418,425),(419,428),(419,445),(420,433),(420,440),(420,496),(421,432),(422,416),(422,426),(423,495),(424,434),(424,495),(425,444),(425,496),(426,423),(426,424),(427,497),(428,124),(428,497),(429,400),(429,490),(430,427),(430,445),(431,421),(433,431),(433,490),(434,445),(434,489),(435,402),(435,492),(436,399),(436,493),(437,423),(437,493),(438,403),(438,494),(439,387),(439,428),(439,489),(440,388),(440,431),(441,392),(442,380),(442,491),(443,385),(443,492),(444,490),(445,497),(446,29),(446,390),(446,464),(447,28),(447,349),(447,386),(448,188),(448,284),(448,478),(449,170),(449,301),(450,168),(450,465),(451,167),(452,169),(453,181),(453,361),(453,484),(454,256),(454,368),(454,467),(455,218),(455,354),(455,368),(456,174),(456,324),(456,360),(457,217),(457,336),(457,466),(458,208),(458,265),(458,403),(459,148),(459,268),(459,491),(460,226),(460,283),(460,472),(461,207),(461,347),(461,356),(462,165),(462,353),(463,184),(464,258),(464,491),(465,141),(466,206),(466,407),(467,254),(467,422),(468,230),(468,364),(469,208),(469,494),(470,188),(470,494),(471,148),(471,380),(472,136),(472,369),(472,455),(473,177),(473,346),(473,374),(474,229),(474,415),(475,223),(476,185),(476,417),(477,147),(478,143),(478,355),(478,402),(479,212),(479,350),(479,399),(480,137),(481,164),(481,489),(482,212),(482,493),(483,205),(483,292),(483,492),(484,142),(484,351),(484,393),(485,146),(485,378),(486,147),(486,292),(487,187),(487,231),(488,146),(488,178),(488,379),(489,487),(489,497),(490,370),(490,421),(491,269),(491,475),(492,396),(492,411),(493,395),(494,355),(494,397),(495,97),(495,489),(496,388),(496,490),(497,187)],498)
=> ? = 6 - 1
[5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The number of posets with the same order polynomial.
The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$.
See sections 3.12 and 3.15 of [1].
Matching statistic: St000525
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000525: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000525: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0 - 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0 - 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,1),(0,2),(1,12),(1,14),(2,11),(2,13),(3,145),(4,146),(5,31),(5,155),(6,32),(6,156),(7,15),(7,143),(8,16),(8,144),(9,37),(9,114),(10,38),(10,115),(11,17),(11,157),(12,18),(12,158),(13,29),(13,35),(13,157),(14,30),(14,36),(14,158),(15,84),(16,85),(17,147),(18,148),(19,102),(19,133),(20,103),(20,134),(21,23),(21,149),(21,153),(22,24),(22,150),(22,154),(23,118),(23,128),(24,119),(24,129),(25,62),(25,82),(26,63),(26,83),(27,74),(27,139),(28,75),(28,140),(29,100),(29,126),(30,101),(30,127),(31,55),(31,135),(32,56),(32,136),(33,47),(33,72),(33,98),(34,48),(34,73),(34,99),(35,100),(35,130),(35,132),(36,101),(36,131),(36,132),(37,96),(37,110),(37,112),(38,97),(38,111),(38,113),(40,201),(40,202),(41,225),(42,226),(43,221),(43,223),(44,222),(44,224),(45,213),(45,221),(46,214),(46,222),(47,213),(47,223),(48,214),(48,224),(49,161),(50,162),(51,183),(51,227),(52,184),(52,228),(53,173),(53,195),(54,174),(54,196),(55,185),(56,186),(57,179),(57,180),(58,175),(58,177),(59,176),(59,178),(60,171),(60,177),(61,172),(61,178),(62,215),(63,216),(64,220),(65,219),(66,219),(67,220),(68,183),(68,203),(69,184),(69,204),(70,179),(70,199),(71,180),(71,200),(72,9),(72,223),(73,10),(73,224),(74,3),(74,189),(75,4),(75,190),(76,53),(76,205),(76,227),(77,54),(77,206),(77,228),(78,53),(78,207),(79,54),(79,208),(80,58),(80,163),(80,167),(81,59),(81,164),(81,168),(82,49),(82,215),(83,50),(83,216),(84,94),(85,95),(86,68),(86,197),(87,69),(87,198),(88,94),(89,95),(90,55),(90,169),(91,56),(91,170),(92,108),(92,170),(93,109),(93,169),(94,39),(95,39),(96,86),(96,163),(97,87),(97,164),(98,19),(98,137),(98,213),(99,20),(99,138),(99,214),(100,21),(100,116),(100,159),(101,22),(101,117),(101,160),(102,90),(102,217),(103,91),(103,218),(104,51),(104,181),(105,52),(105,182),(106,41),(106,187),(107,42),(107,188),(108,40),(108,175),(108,194),(109,40),(109,176),(109,193),(110,60),(110,163),(110,191),(111,61),(111,164),(111,192),(112,104),(112,191),(113,105),(113,192),(114,112),(115,113),(116,124),(116,149),(117,124),(117,150),(118,91),(118,92),(119,90),(119,93),(120,58),(120,60),(120,225),(121,59),(121,61),(121,226),(122,92),(122,125),(122,218),(123,93),(123,125),(123,217),(124,122),(124,123),(125,108),(125,109),(125,167),(125,168),(126,43),(126,72),(126,159),(127,44),(127,73),(127,160),(128,41),(128,120),(129,42),(129,121),(130,45),(130,98),(130,159),(131,46),(131,99),(131,160),(132,116),(132,117),(133,80),(133,96),(133,217),(134,81),(134,97),(134,218),(135,51),(135,76),(135,185),(136,52),(136,77),(136,186),(137,133),(137,151),(137,165),(138,134),(138,152),(138,166),(139,57),(139,70),(139,189),(140,57),(140,71),(140,190),(141,76),(141,78),(141,181),(142,77),(142,79),(142,182),(143,84),(143,88),(144,85),(144,89),(145,64),(145,67),(146,65),(146,66),(147,43),(147,45),(147,47),(148,44),(148,46),(148,48),(149,103),(149,118),(149,122),(150,102),(150,119),(150,123),(151,80),(151,110),(151,120),(151,187),(152,81),(152,111),(152,121),(152,188),(153,106),(153,128),(153,151),(154,107),(154,129),(154,152),(155,104),(155,135),(155,141),(156,105),(156,136),(156,142),(157,33),(157,126),(157,130),(157,147),(158,34),(158,127),(158,131),(158,148),(159,137),(159,153),(159,221),(160,138),(160,154),(160,222),(161,143),(162,144),(163,27),(163,177),(163,197),(164,28),(164,178),(164,198),(165,155),(165,187),(166,156),(166,188),(167,175),(167,193),(167,197),(168,176),(168,194),(168,198),(169,68),(169,185),(169,193),(170,69),(170,186),(170,194),(171,207),(172,208),(173,215),(173,231),(174,216),(174,232),(175,201),(175,209),(176,202),(176,210),(177,74),(177,209),(178,75),(178,210),(179,66),(179,233),(180,67),(180,233),(181,25),(181,207),(181,227),(182,26),(182,208),(182,228),(183,229),(184,230),(185,183),(185,205),(186,184),(186,206),(187,141),(187,191),(187,225),(188,142),(188,192),(188,226),(189,145),(189,180),(189,199),(190,146),(190,179),(190,200),(191,171),(191,181),(192,172),(192,182),(193,201),(193,203),(193,205),(194,202),(194,204),(194,206),(195,231),(196,232),(197,139),(197,203),(197,209),(198,140),(198,204),(198,210),(199,64),(199,233),(200,65),(200,233),(201,195),(201,211),(202,196),(202,212),(203,70),(203,211),(203,229),(204,71),(204,212),(204,230),(205,195),(205,229),(206,196),(206,230),(207,62),(207,173),(208,63),(208,174),(209,189),(209,211),(210,190),(210,212),(211,199),(211,231),(212,200),(212,232),(213,5),(213,165),(214,6),(214,166),(215,7),(215,161),(216,8),(216,162),(217,86),(217,167),(217,169),(218,87),(218,168),(218,170),(219,88),(220,89),(221,106),(221,165),(222,107),(222,166),(223,114),(224,115),(225,78),(225,171),(226,79),(226,172),(227,82),(227,173),(227,229),(228,83),(228,174),(228,230),(229,49),(229,231),(230,50),(230,232),(231,161),(232,162),(233,219),(233,220)],234)
=> ? = 4 - 1
[4,5,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,18),(0,19),(1,17),(1,50),(2,10),(2,13),(3,63),(3,272),(4,284),(5,283),(6,64),(6,287),(7,8),(7,325),(8,309),(9,60),(9,314),(10,38),(10,310),(11,40),(11,149),(12,49),(12,59),(12,321),(13,22),(13,58),(13,310),(14,43),(14,278),(15,41),(15,280),(16,42),(16,313),(17,34),(17,35),(17,304),(18,1),(18,184),(19,12),(19,37),(19,184),(20,231),(20,235),(21,233),(21,298),(22,238),(22,294),(23,68),(23,318),(23,324),(24,123),(24,241),(25,226),(25,237),(26,52),(26,229),(26,322),(27,124),(27,230),(28,127),(28,302),(29,125),(29,303),(30,120),(30,296),(31,232),(31,240),(32,225),(32,293),(33,224),(33,292),(34,243),(34,295),(35,243),(35,316),(36,227),(36,289),(37,51),(37,290),(37,321),(38,242),(38,320),(39,219),(39,300),(40,222),(40,223),(41,220),(41,221),(42,121),(42,297),(43,122),(43,228),(44,66),(44,236),(44,291),(45,65),(45,126),(45,234),(46,67),(46,288),(46,299),(47,54),(47,301),(47,317),(48,55),(48,319),(48,323),(49,53),(49,239),(49,315),(50,242),(50,304),(51,167),(51,308),(52,206),(52,273),(53,255),(53,263),(54,157),(54,253),(55,254),(55,266),(56,181),(56,183),(56,218),(57,182),(57,267),(57,270),(58,238),(58,269),(58,271),(59,239),(59,268),(59,271),(60,106),(60,180),(60,274),(61,89),(61,215),(61,217),(62,90),(62,119),(62,216),(63,145),(63,189),(63,194),(64,57),(64,255),(64,306),(64,316),(65,155),(65,195),(65,197),(66,153),(66,154),(66,156),(67,101),(67,196),(67,262),(68,131),(68,244),(68,245),(69,452),(70,466),(71,417),(72,454),(72,456),(73,451),(73,453),(74,426),(74,449),(75,425),(75,455),(76,414),(76,416),(77,336),(77,456),(78,336),(78,454),(79,335),(79,344),(80,342),(80,450),(81,343),(81,453),(82,11),(83,415),(83,467),(84,413),(84,467),(85,455),(85,465),(86,341),(86,464),(87,472),(88,471),(89,342),(89,441),(90,3),(90,451),(91,357),(92,358),(93,358),(94,394),(94,406),(95,364),(95,452),(96,372),(96,468),(97,371),(97,457),(98,364),(98,424),(99,359),(99,404),(100,376),(100,400),(101,326),(101,377),(102,332),(102,363),(103,331),(103,351),(104,332),(104,374),(105,348),(105,463),(106,341),(106,375),(107,350),(108,409),(109,410),(110,459),(111,459),(112,421),(112,468),(113,458),(114,397),(115,356),(115,408),(116,371),(116,418),(117,372),(117,420),(118,370),(118,411),(119,36),(119,343),(119,451),(120,432),(121,393),(122,369),(123,391),(124,4),(124,392),(125,5),(125,390),(126,46),(126,368),(127,39),(127,365),(128,99),(129,150),(130,149),(131,264),(131,461),(132,178),(132,470),(133,212),(133,462),(134,218),(134,447),(134,449),(135,138),(136,198),(136,436),(137,91),(137,458),(138,92),(138,422),(139,99),(139,407),(139,465),(140,202),(140,464),(141,118),(141,469),(142,97),(142,435),(143,188),(144,100),(144,419),(144,457),(145,200),(145,433),(146,102),(146,345),(146,433),(147,103),(147,346),(147,434),(148,126),(148,344),(149,223),(150,92),(151,193),(152,122),(152,423),(153,163),(153,338),(154,209),(154,337),(155,201),(155,434),(156,240),(156,337),(156,338),(157,265),(158,221),(158,471),(159,93),(160,93),(161,181),(161,444),(161,447),(162,94),(162,405),(162,437),(163,180),(163,438),(164,130),(164,438),(165,175),(165,460),(166,176),(166,337),(166,460),(167,252),(168,106),(168,340),(169,121),(169,339),(170,165),(170,442),(171,210),(171,340),(172,211),(172,339),(173,98),(173,328),(174,79),(174,441),(174,450),(175,78),(175,329),(175,443),(176,77),(176,440),(176,443),(177,69),(177,439),(178,71),(178,328),(179,74),(179,335),(179,432),(180,199),(180,341),(181,288),(181,329),(182,131),(182,327),(182,442),(183,72),(183,329),(183,440),(184,6),(184,290),(185,231),(185,355),(186,203),(186,334),(186,385),(187,105),(187,388),(188,214),(188,382),(189,136),(189,381),(190,213),(190,385),(191,100),(191,389),(192,94),(192,378),(193,108),(193,357),(194,104),(194,381),(194,433),(195,142),(195,384),(196,188),(196,377),(197,205),(197,384),(197,434),(198,222),(198,356),(198,437),(199,186),(199,383),(200,115),(200,367),(201,116),(201,366),(202,109),(202,349),(203,95),(203,380),(203,439),(204,207),(204,382),(205,101),(205,331),(205,395),(206,71),(206,352),(207,83),(207,360),(207,403),(208,70),(208,387),(209,86),(209,362),(210,76),(210,351),(210,401),(211,76),(211,363),(211,402),(212,75),(212,326),(212,349),(213,98),(213,354),(213,380),(214,84),(214,359),(214,360),(215,33),(215,286),(215,342),(216,32),(216,285),(216,343),(217,45),(217,148),(217,441),(218,21),(218,247),(218,440),(219,151),(220,150),(221,159),(222,114),(222,334),(223,177),(223,334),(224,169),(224,446),(225,168),(225,445),(226,143),(227,82),(228,138),(228,369),(229,206),(229,396),(230,107),(230,392),(231,91),(231,373),(232,130),(232,353),(233,132),(233,361),(234,195),(234,368),(235,152),(235,373),(236,31),(236,156),(236,472),(237,26),(237,250),(238,48),(238,249),(238,330),(239,23),(239,248),(239,333),(240,47),(240,276),(240,353),(241,20),(241,185),(241,391),(242,261),(243,7),(243,307),(244,105),(244,256),(244,461),(245,168),(245,171),(246,102),(246,104),(246,466),(247,212),(247,298),(247,347),(248,259),(248,318),(249,259),(249,319),(250,204),(250,322),(251,173),(251,213),(251,470),(252,120),(252,179),(253,173),(253,178),(254,169),(254,172),(255,170),(255,182),(255,448),(256,103),(256,205),(256,463),(257,172),(257,260),(257,446),(258,171),(258,260),(258,445),(259,257),(259,258),(260,210),(260,211),(260,345),(260,346),(261,87),(261,291),(262,85),(262,139),(262,326),(263,79),(263,148),(263,448),(264,86),(264,140),(265,69),(265,95),(266,70),(266,246),(267,154),(267,166),(267,442),(268,80),(268,215),(268,333),(269,81),(269,216),(269,330),(270,74),(270,134),(270,327),(271,248),(271,249),(272,82),(272,189),(273,83),(273,84),(273,352),(274,162),(274,198),(274,375),(275,162),(275,192),(275,436),(276,282),(276,317),(277,144),(277,191),(277,435),(278,135),(278,228),(279,128),(279,139),(280,129),(280,220),(281,186),(281,305),(282,190),(282,251),(283,110),(283,111),(284,88),(284,158),(285,293),(285,312),(285,379),(286,292),(286,311),(286,386),(287,30),(287,252),(287,306),(288,262),(288,279),(289,163),(289,164),(290,167),(290,287),(291,153),(291,289),(291,472),(292,147),(292,155),(292,446),(293,145),(293,146),(293,445),(294,73),(294,90),(294,330),(295,87),(295,236),(296,56),(296,134),(296,161),(296,432),(297,97),(297,144),(297,393),(298,75),(298,85),(298,361),(299,143),(299,196),(300,113),(300,137),(301,157),(301,305),(302,96),(302,117),(302,365),(303,96),(303,112),(303,390),(304,44),(304,261),(304,295),(305,177),(305,203),(305,265),(306,179),(306,270),(306,296),(306,448),(307,165),(307,166),(307,325),(308,80),(308,89),(308,174),(309,72),(309,77),(309,78),(310,62),(310,269),(310,294),(310,320),(311,147),(311,197),(311,256),(311,388),(312,146),(312,194),(312,246),(312,387),(313,142),(313,277),(313,297),(314,136),(314,274),(314,275),(315,174),(315,217),(315,263),(315,333),(316,170),(316,267),(316,307),(317,132),(317,251),(317,253),(318,225),(318,245),(318,258),(319,224),(319,254),(319,257),(320,73),(320,81),(320,119),(321,61),(321,268),(321,308),(321,315),(322,207),(322,214),(322,273),(322,396),(323,208),(323,266),(323,312),(324,187),(324,244),(324,311),(325,175),(325,176),(325,183),(325,309),(326,407),(326,455),(327,426),(327,447),(327,461),(328,118),(328,417),(328,424),(329,279),(329,454),(330,285),(330,323),(330,453),(331,127),(331,431),(332,125),(332,427),(333,286),(333,324),(333,450),(334,397),(334,439),(335,426),(335,444),(336,109),(336,399),(337,276),(337,362),(338,353),(338,438),(339,116),(339,393),(339,402),(340,115),(340,375),(340,401),(341,383),(342,16),(342,386),(343,9),(343,379),(344,368),(345,363),(345,367),(345,401),(346,351),(346,366),(346,402),(347,349),(347,361),(347,399),(348,462),(349,407),(349,410),(349,425),(350,280),(351,416),(351,431),(352,413),(352,415),(352,417),(353,281),(353,301),(354,424),(354,469),(355,278),(356,114),(356,482),(357,409),(359,413),(359,479),(360,284),(360,467),(360,479),(361,425),(361,465),(361,470),(362,282),(362,464),(363,414),(363,427),(364,398),(365,300),(365,420),(365,468),(366,302),(366,418),(366,431),(367,303),(367,408),(367,427),(368,299),(369,422),(370,108),(370,481),(371,476),(372,110),(372,475),(373,357),(373,423),(374,378),(375,356),(375,405),(376,392),(376,478),(377,382),(378,123),(378,394),(379,314),(379,387),(380,364),(380,412),(381,374),(381,436),(382,360),(383,385),(384,395),(384,435),(385,380),(385,397),(386,313),(386,388),(387,275),(387,381),(387,466),(388,277),(388,384),(388,463),(389,124),(389,376),(390,283),(390,372),(390,421),(391,14),(391,355),(392,15),(392,350),(393,371),(393,419),(394,391),(394,477),(395,377),(395,389),(396,352),(396,359),(396,403),(397,412),(398,423),(399,396),(399,410),(400,478),(401,405),(401,408),(401,414),(402,416),(402,418),(402,419),(403,415),(403,479),(404,479),(405,406),(405,482),(406,477),(407,404),(407,480),(408,112),(408,429),(408,482),(409,422),(410,403),(410,404),(411,481),(412,373),(412,398),(413,411),(413,473),(414,406),(414,429),(415,430),(415,473),(416,400),(416,428),(417,411),(417,430),(418,117),(418,428),(418,476),(419,400),(419,476),(420,113),(420,475),(421,111),(421,475),(422,358),(423,369),(423,409),(424,370),(424,430),(425,354),(425,480),(426,348),(426,474),(427,390),(427,429),(428,420),(428,478),(429,421),(429,477),(430,481),(431,365),(431,428),(432,25),(432,444),(432,449),(433,29),(433,332),(433,367),(434,28),(434,331),(434,366),(435,27),(435,389),(435,457),(436,24),(436,378),(436,437),(437,241),(437,394),(437,482),(438,199),(438,281),(439,235),(439,412),(439,452),(440,233),(440,347),(440,456),(441,234),(441,344),(442,209),(442,264),(442,460),(443,202),(443,336),(443,347),(444,226),(444,474),(445,200),(445,340),(445,345),(446,201),(446,339),(446,346),(447,133),(447,247),(447,474),(448,161),(448,327),(448,335),(449,237),(449,474),(450,187),(450,386),(451,227),(451,272),(452,152),(452,398),(453,208),(453,379),(454,128),(455,219),(455,480),(456,229),(456,399),(457,230),(457,376),(457,476),(458,135),(459,129),(460,140),(460,362),(460,443),(461,133),(461,348),(462,204),(463,191),(463,395),(464,190),(464,383),(465,141),(465,480),(466,192),(466,374),(467,158),(467,473),(468,137),(468,475),(469,193),(469,370),(470,141),(470,328),(470,354),(471,159),(471,160),(472,164),(472,232),(472,338),(473,471),(473,481),(474,250),(474,462),(475,458),(475,459),(476,107),(476,478),(477,355),(478,350),(479,88),(479,473),(480,151),(480,469),(481,160),(482,185),(482,477)],483)
=> ? = 6 - 1
[5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,23),(0,24),(1,14),(2,13),(3,35),(3,78),(4,36),(4,79),(5,33),(5,54),(6,34),(6,55),(7,15),(7,80),(8,16),(8,81),(9,17),(9,52),(10,18),(10,53),(11,19),(11,31),(11,82),(12,20),(12,32),(12,83),(13,76),(14,77),(15,58),(16,59),(17,91),(18,92),(19,21),(19,74),(20,22),(20,75),(21,27),(21,88),(22,28),(22,89),(23,11),(23,90),(24,12),(24,90),(25,68),(25,101),(26,69),(26,102),(27,66),(27,97),(28,67),(28,98),(29,47),(29,99),(30,46),(30,100),(31,56),(31,74),(32,57),(32,75),(33,30),(33,97),(33,104),(34,29),(34,98),(34,104),(35,25),(35,84),(35,86),(36,26),(36,85),(36,87),(38,121),(38,122),(39,107),(39,123),(40,108),(40,124),(41,107),(41,108),(42,125),(43,126),(44,116),(44,121),(45,115),(45,122),(46,117),(47,118),(48,125),(49,126),(50,113),(51,114),(52,2),(53,1),(54,3),(55,4),(56,54),(57,55),(58,52),(59,53),(60,50),(60,119),(61,51),(61,120),(62,76),(62,119),(63,77),(63,120),(64,37),(65,37),(66,78),(67,79),(68,62),(68,105),(69,63),(69,106),(70,42),(70,111),(71,43),(71,112),(72,44),(72,110),(73,45),(73,109),(74,7),(74,88),(75,8),(75,89),(76,93),(77,94),(78,84),(79,85),(80,9),(80,58),(81,10),(81,59),(82,5),(82,56),(83,6),(83,57),(84,68),(84,91),(84,123),(85,69),(85,92),(85,124),(86,73),(86,101),(86,123),(87,72),(87,102),(87,124),(88,66),(88,80),(89,67),(89,81),(90,82),(90,83),(91,60),(91,62),(92,61),(92,63),(93,42),(93,48),(94,43),(94,49),(95,39),(95,41),(95,117),(96,40),(96,41),(96,118),(97,46),(97,95),(98,47),(98,96),(99,39),(99,86),(99,118),(100,40),(100,87),(100,117),(101,38),(101,45),(101,105),(102,38),(102,44),(102,106),(103,64),(103,65),(104,95),(104,96),(104,99),(104,100),(105,115),(105,119),(105,121),(106,116),(106,120),(106,122),(107,109),(108,110),(109,50),(109,115),(110,51),(110,116),(111,103),(111,125),(112,103),(112,126),(113,48),(113,111),(114,49),(114,112),(115,113),(115,127),(116,114),(116,127),(117,72),(117,108),(118,73),(118,107),(119,70),(119,93),(119,113),(120,71),(120,94),(120,114),(121,70),(121,127),(122,71),(122,127),(123,60),(123,105),(123,109),(124,61),(124,106),(124,110),(125,64),(126,65),(127,111),(127,112)],128)
=> ? = 4 - 1
[5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,25),(0,26),(0,66),(1,56),(2,55),(3,7),(3,200),(4,168),(5,303),(6,260),(7,264),(8,70),(8,306),(9,10),(9,68),(9,201),(10,18),(10,307),(11,48),(11,203),(12,47),(12,337),(13,52),(13,304),(14,53),(14,289),(15,46),(15,240),(16,44),(16,98),(17,54),(17,202),(18,45),(18,335),(19,51),(19,336),(20,24),(20,50),(20,204),(21,49),(21,344),(22,36),(22,338),(23,37),(23,38),(23,341),(24,34),(24,43),(24,253),(25,9),(25,276),(26,23),(26,40),(26,163),(27,256),(27,325),(28,130),(28,131),(29,76),(29,258),(29,327),(30,127),(30,323),(31,62),(31,129),(31,324),(32,75),(32,319),(32,328),(33,310),(33,326),(34,259),(34,342),(35,77),(35,245),(35,318),(36,242),(36,314),(37,133),(37,322),(38,64),(38,133),(38,339),(39,74),(39,254),(39,343),(40,73),(40,315),(40,341),(41,60),(41,257),(41,320),(42,59),(42,132),(42,329),(43,57),(43,259),(43,312),(44,63),(44,316),(44,345),(45,321),(45,330),(46,65),(46,330),(46,345),(47,251),(47,309),(48,246),(48,252),(49,241),(49,340),(50,249),(50,253),(51,250),(51,313),(52,244),(52,247),(53,126),(53,248),(54,61),(54,311),(54,317),(55,128),(55,255),(56,58),(56,243),(56,308),(57,162),(57,316),(58,216),(58,293),(59,214),(59,215),(60,159),(60,161),(61,160),(61,286),(62,140),(62,261),(63,155),(63,273),(64,274),(64,287),(65,96),(65,282),(66,20),(66,163),(66,276),(67,196),(67,198),(67,239),(68,197),(68,199),(68,307),(69,233),(69,234),(69,288),(70,72),(70,287),(70,334),(70,342),(71,125),(71,195),(71,238),(72,157),(72,267),(72,280),(73,173),(73,199),(73,333),(74,213),(74,236),(74,295),(75,102),(75,210),(75,262),(76,94),(76,150),(76,277),(77,69),(77,279),(77,291),(77,294),(78,432),(79,463),(80,465),(81,430),(82,366),(82,466),(83,367),(84,442),(84,464),(85,434),(85,481),(86,431),(87,480),(88,441),(89,438),(89,470),(90,429),(90,433),(91,364),(91,473),(92,354),(92,467),(93,354),(93,462),(94,353),(94,365),(95,469),(95,470),(96,468),(97,487),(98,17),(99,359),(99,385),(100,382),(101,387),(101,425),(102,362),(102,404),(103,394),(103,436),(104,410),(104,435),(105,384),(105,419),(106,376),(106,423),(107,380),(107,474),(108,364),(108,389),(109,394),(109,482),(110,384),(110,476),(111,356),(111,401),(112,389),(112,453),(113,14),(113,391),(114,375),(114,387),(115,351),(116,427),(116,428),(117,426),(118,476),(119,488),(120,403),(121,381),(122,391),(123,385),(123,402),(124,388),(125,33),(125,357),(125,366),(126,352),(127,6),(127,446),(128,400),(129,13),(129,373),(130,5),(130,392),(131,4),(131,392),(132,32),(132,409),(133,19),(133,457),(134,191),(134,352),(135,153),(135,479),(136,218),(136,480),(137,232),(138,239),(138,459),(138,464),(139,171),(139,460),(140,175),(140,485),(141,128),(142,123),(142,478),(143,121),(143,463),(144,114),(144,439),(144,481),(145,100),(145,465),(146,99),(146,486),(147,237),(148,222),(148,475),(149,158),(149,447),(150,180),(150,353),(151,104),(151,351),(151,365),(152,111),(152,357),(152,466),(153,105),(153,350),(154,209),(154,474),(155,241),(155,363),(156,243),(157,250),(157,347),(157,460),(158,126),(158,349),(159,166),(159,360),(160,176),(160,346),(161,129),(161,360),(162,202),(163,8),(163,315),(164,255),(164,487),(165,106),(165,452),(166,242),(166,449),(167,318),(168,141),(169,101),(169,418),(170,134),(171,189),(171,472),(172,145),(172,450),(173,272),(173,451),(174,270),(174,449),(175,156),(176,251),(176,458),(177,120),(177,458),(178,99),(178,443),(178,483),(179,106),(179,437),(179,482),(180,308),(180,452),(181,142),(181,448),(182,190),(182,358),(182,472),(183,94),(183,462),(183,467),(184,86),(185,86),(185,440),(186,88),(186,349),(187,78),(188,79),(188,355),(189,93),(189,348),(189,455),(190,92),(190,454),(190,455),(191,80),(191,450),(192,84),(192,356),(192,446),(193,81),(193,350),(194,83),(194,456),(195,217),(195,366),(196,319),(196,348),(197,240),(197,451),(198,183),(198,348),(198,454),(199,266),(199,451),(200,12),(200,297),(201,15),(201,197),(202,317),(203,3),(203,246),(204,16),(204,249),(205,228),(205,396),(206,107),(206,408),(207,196),(207,390),(207,459),(208,110),(208,397),(209,102),(209,415),(210,221),(210,404),(211,104),(211,393),(211,469),(212,116),(212,395),(213,205),(213,414),(214,149),(214,406),(215,209),(215,406),(216,100),(216,412),(217,290),(217,407),(218,117),(218,371),(219,220),(219,379),(220,213),(220,410),(220,483),(221,235),(221,398),(222,103),(222,362),(222,371),(223,227),(223,398),(224,132),(224,401),(225,244),(225,378),(225,448),(226,87),(226,369),(227,85),(227,377),(227,424),(228,90),(228,375),(228,420),(229,88),(229,386),(230,89),(230,374),(231,78),(231,370),(232,81),(232,399),(233,225),(233,361),(233,413),(234,219),(234,413),(235,144),(235,376),(235,377),(236,105),(236,383),(236,414),(237,90),(237,381),(237,382),(238,42),(238,224),(238,357),(239,27),(239,268),(239,454),(240,35),(240,167),(241,300),(242,170),(243,216),(243,359),(244,284),(244,477),(245,294),(245,367),(246,200),(246,358),(247,285),(247,477),(248,172),(248,352),(249,98),(250,283),(250,471),(251,265),(252,226),(252,358),(253,11),(253,312),(254,236),(254,416),(255,231),(255,400),(256,135),(256,372),(257,31),(257,161),(257,456),(258,39),(258,269),(259,21),(259,332),(260,119),(260,278),(261,181),(261,225),(261,485),(262,109),(262,179),(262,362),(263,193),(263,232),(263,479),(264,176),(264,177),(265,110),(265,118),(266,194),(266,320),(267,182),(267,252),(267,460),(268,222),(268,325),(268,368),(269,223),(269,343),(270,175),(270,289),(271,180),(271,221),(272,127),(272,192),(273,112),(273,279),(273,363),(274,111),(274,224),(274,461),(275,108),(275,112),(275,468),(276,201),(276,204),(277,178),(277,220),(277,365),(278,219),(278,277),(278,488),(279,288),(279,298),(279,453),(280,84),(280,138),(280,347),(281,91),(281,108),(281,363),(282,91),(282,311),(282,468),(283,87),(283,136),(284,79),(284,143),(285,80),(285,145),(286,89),(286,95),(286,346),(287,139),(287,157),(287,461),(288,95),(288,211),(288,361),(289,156),(289,248),(290,154),(290,215),(290,408),(291,234),(291,278),(291,405),(292,228),(292,237),(292,411),(293,101),(293,114),(293,412),(294,233),(294,261),(294,405),(294,453),(295,85),(295,144),(295,383),(296,165),(296,179),(297,302),(297,337),(298,151),(298,211),(298,484),(299,186),(299,229),(299,447),(300,115),(300,151),(301,134),(301,331),(302,137),(302,263),(303,97),(303,164),(304,247),(304,331),(305,184),(305,185),(306,30),(306,272),(306,334),(307,41),(307,266),(307,335),(308,169),(308,293),(308,359),(309,153),(309,193),(310,166),(310,174),(311,230),(311,286),(311,473),(312,162),(312,203),(313,107),(313,154),(313,471),(314,158),(314,186),(315,173),(315,306),(316,155),(316,281),(317,160),(317,264),(317,473),(318,260),(318,291),(318,367),(319,262),(319,296),(320,159),(320,310),(320,456),(321,83),(321,245),(322,82),(322,195),(322,457),(323,67),(323,138),(323,207),(323,446),(324,140),(324,270),(324,373),(325,103),(325,109),(325,372),(326,113),(326,122),(327,150),(327,271),(328,210),(328,271),(329,113),(329,214),(329,409),(330,96),(330,275),(331,172),(331,191),(331,285),(332,171),(332,182),(332,344),(333,82),(333,125),(333,152),(334,192),(334,280),(334,323),(334,461),(335,194),(335,257),(335,321),(336,206),(336,290),(336,313),(337,135),(337,263),(337,309),(338,149),(338,299),(338,314),(339,152),(339,238),(339,274),(339,457),(340,92),(340,93),(340,183),(341,71),(341,322),(341,333),(341,339),(342,139),(342,267),(342,332),(343,227),(343,235),(343,295),(343,416),(344,189),(344,190),(344,198),(344,340),(345,273),(345,275),(345,281),(345,282),(346,438),(346,458),(346,469),(347,442),(347,459),(347,471),(348,296),(348,462),(349,131),(349,441),(350,116),(350,419),(350,430),(351,123),(351,435),(351,443),(352,450),(353,452),(354,117),(354,422),(355,305),(355,463),(356,390),(356,442),(357,326),(357,329),(357,401),(358,297),(358,369),(359,412),(359,418),(360,373),(360,449),(361,393),(361,448),(361,470),(362,394),(362,437),(363,298),(363,300),(363,389),(364,374),(365,410),(365,443),(366,22),(366,407),(367,119),(367,405),(368,371),(368,372),(368,422),(369,302),(369,480),(370,432),(371,426),(371,436),(371,437),(372,436),(372,479),(372,482),(373,301),(373,304),(374,120),(374,438),(375,429),(375,496),(376,439),(376,495),(377,303),(377,481),(377,495),(378,477),(378,486),(379,483),(379,486),(380,475),(381,433),(381,444),(382,429),(382,444),(383,419),(383,434),(383,439),(384,417),(385,418),(386,130),(386,441),(387,124),(387,496),(388,421),(389,115),(389,484),(390,327),(390,491),(391,1),(392,2),(393,435),(393,478),(393,494),(394,493),(395,427),(396,417),(396,420),(397,305),(397,476),(398,377),(399,395),(399,430),(400,370),(401,122),(401,409),(402,121),(402,411),(403,118),(403,397),(404,398),(405,413),(405,485),(405,488),(406,415),(406,447),(407,338),(407,408),(408,299),(408,406),(408,474),(409,328),(409,391),(410,414),(410,492),(411,381),(411,420),(412,375),(412,382),(412,425),(413,378),(413,379),(414,384),(414,396),(415,386),(415,404),(416,376),(416,383),(416,424),(417,440),(418,425),(419,428),(419,445),(420,433),(420,440),(420,496),(421,432),(422,416),(422,426),(423,495),(424,434),(424,495),(425,444),(425,496),(426,423),(426,424),(427,497),(428,124),(428,497),(429,400),(429,490),(430,427),(430,445),(431,421),(433,431),(433,490),(434,445),(434,489),(435,402),(435,492),(436,399),(436,493),(437,423),(437,493),(438,403),(438,494),(439,387),(439,428),(439,489),(440,388),(440,431),(441,392),(442,380),(442,491),(443,385),(443,492),(444,490),(445,497),(446,29),(446,390),(446,464),(447,28),(447,349),(447,386),(448,188),(448,284),(448,478),(449,170),(449,301),(450,168),(450,465),(451,167),(452,169),(453,181),(453,361),(453,484),(454,256),(454,368),(454,467),(455,218),(455,354),(455,368),(456,174),(456,324),(456,360),(457,217),(457,336),(457,466),(458,208),(458,265),(458,403),(459,148),(459,268),(459,491),(460,226),(460,283),(460,472),(461,207),(461,347),(461,356),(462,165),(462,353),(463,184),(464,258),(464,491),(465,141),(466,206),(466,407),(467,254),(467,422),(468,230),(468,364),(469,208),(469,494),(470,188),(470,494),(471,148),(471,380),(472,136),(472,369),(472,455),(473,177),(473,346),(473,374),(474,229),(474,415),(475,223),(476,185),(476,417),(477,147),(478,143),(478,355),(478,402),(479,212),(479,350),(479,399),(480,137),(481,164),(481,489),(482,212),(482,493),(483,205),(483,292),(483,492),(484,142),(484,351),(484,393),(485,146),(485,378),(486,147),(486,292),(487,187),(487,231),(488,146),(488,178),(488,379),(489,487),(489,497),(490,370),(490,421),(491,269),(491,475),(492,396),(492,411),(493,395),(494,355),(494,397),(495,97),(495,489),(496,388),(496,490),(497,187)],498)
=> ? = 6 - 1
[5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The number of posets with the same zeta polynomial.
The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset.
See section 3.12 of [1].
Since
$$
Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k,
$$
where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.
Matching statistic: St000526
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000526: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000526: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 29%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0 - 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0 - 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 0 - 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 0 - 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 4 - 1
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 6 - 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? = 2 - 1
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 2 - 1
[4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? = 4 - 1
[4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? = 4 - 1
[4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 6 - 1
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 2 - 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,1),(0,2),(1,12),(1,14),(2,11),(2,13),(3,145),(4,146),(5,31),(5,155),(6,32),(6,156),(7,15),(7,143),(8,16),(8,144),(9,37),(9,114),(10,38),(10,115),(11,17),(11,157),(12,18),(12,158),(13,29),(13,35),(13,157),(14,30),(14,36),(14,158),(15,84),(16,85),(17,147),(18,148),(19,102),(19,133),(20,103),(20,134),(21,23),(21,149),(21,153),(22,24),(22,150),(22,154),(23,118),(23,128),(24,119),(24,129),(25,62),(25,82),(26,63),(26,83),(27,74),(27,139),(28,75),(28,140),(29,100),(29,126),(30,101),(30,127),(31,55),(31,135),(32,56),(32,136),(33,47),(33,72),(33,98),(34,48),(34,73),(34,99),(35,100),(35,130),(35,132),(36,101),(36,131),(36,132),(37,96),(37,110),(37,112),(38,97),(38,111),(38,113),(40,201),(40,202),(41,225),(42,226),(43,221),(43,223),(44,222),(44,224),(45,213),(45,221),(46,214),(46,222),(47,213),(47,223),(48,214),(48,224),(49,161),(50,162),(51,183),(51,227),(52,184),(52,228),(53,173),(53,195),(54,174),(54,196),(55,185),(56,186),(57,179),(57,180),(58,175),(58,177),(59,176),(59,178),(60,171),(60,177),(61,172),(61,178),(62,215),(63,216),(64,220),(65,219),(66,219),(67,220),(68,183),(68,203),(69,184),(69,204),(70,179),(70,199),(71,180),(71,200),(72,9),(72,223),(73,10),(73,224),(74,3),(74,189),(75,4),(75,190),(76,53),(76,205),(76,227),(77,54),(77,206),(77,228),(78,53),(78,207),(79,54),(79,208),(80,58),(80,163),(80,167),(81,59),(81,164),(81,168),(82,49),(82,215),(83,50),(83,216),(84,94),(85,95),(86,68),(86,197),(87,69),(87,198),(88,94),(89,95),(90,55),(90,169),(91,56),(91,170),(92,108),(92,170),(93,109),(93,169),(94,39),(95,39),(96,86),(96,163),(97,87),(97,164),(98,19),(98,137),(98,213),(99,20),(99,138),(99,214),(100,21),(100,116),(100,159),(101,22),(101,117),(101,160),(102,90),(102,217),(103,91),(103,218),(104,51),(104,181),(105,52),(105,182),(106,41),(106,187),(107,42),(107,188),(108,40),(108,175),(108,194),(109,40),(109,176),(109,193),(110,60),(110,163),(110,191),(111,61),(111,164),(111,192),(112,104),(112,191),(113,105),(113,192),(114,112),(115,113),(116,124),(116,149),(117,124),(117,150),(118,91),(118,92),(119,90),(119,93),(120,58),(120,60),(120,225),(121,59),(121,61),(121,226),(122,92),(122,125),(122,218),(123,93),(123,125),(123,217),(124,122),(124,123),(125,108),(125,109),(125,167),(125,168),(126,43),(126,72),(126,159),(127,44),(127,73),(127,160),(128,41),(128,120),(129,42),(129,121),(130,45),(130,98),(130,159),(131,46),(131,99),(131,160),(132,116),(132,117),(133,80),(133,96),(133,217),(134,81),(134,97),(134,218),(135,51),(135,76),(135,185),(136,52),(136,77),(136,186),(137,133),(137,151),(137,165),(138,134),(138,152),(138,166),(139,57),(139,70),(139,189),(140,57),(140,71),(140,190),(141,76),(141,78),(141,181),(142,77),(142,79),(142,182),(143,84),(143,88),(144,85),(144,89),(145,64),(145,67),(146,65),(146,66),(147,43),(147,45),(147,47),(148,44),(148,46),(148,48),(149,103),(149,118),(149,122),(150,102),(150,119),(150,123),(151,80),(151,110),(151,120),(151,187),(152,81),(152,111),(152,121),(152,188),(153,106),(153,128),(153,151),(154,107),(154,129),(154,152),(155,104),(155,135),(155,141),(156,105),(156,136),(156,142),(157,33),(157,126),(157,130),(157,147),(158,34),(158,127),(158,131),(158,148),(159,137),(159,153),(159,221),(160,138),(160,154),(160,222),(161,143),(162,144),(163,27),(163,177),(163,197),(164,28),(164,178),(164,198),(165,155),(165,187),(166,156),(166,188),(167,175),(167,193),(167,197),(168,176),(168,194),(168,198),(169,68),(169,185),(169,193),(170,69),(170,186),(170,194),(171,207),(172,208),(173,215),(173,231),(174,216),(174,232),(175,201),(175,209),(176,202),(176,210),(177,74),(177,209),(178,75),(178,210),(179,66),(179,233),(180,67),(180,233),(181,25),(181,207),(181,227),(182,26),(182,208),(182,228),(183,229),(184,230),(185,183),(185,205),(186,184),(186,206),(187,141),(187,191),(187,225),(188,142),(188,192),(188,226),(189,145),(189,180),(189,199),(190,146),(190,179),(190,200),(191,171),(191,181),(192,172),(192,182),(193,201),(193,203),(193,205),(194,202),(194,204),(194,206),(195,231),(196,232),(197,139),(197,203),(197,209),(198,140),(198,204),(198,210),(199,64),(199,233),(200,65),(200,233),(201,195),(201,211),(202,196),(202,212),(203,70),(203,211),(203,229),(204,71),(204,212),(204,230),(205,195),(205,229),(206,196),(206,230),(207,62),(207,173),(208,63),(208,174),(209,189),(209,211),(210,190),(210,212),(211,199),(211,231),(212,200),(212,232),(213,5),(213,165),(214,6),(214,166),(215,7),(215,161),(216,8),(216,162),(217,86),(217,167),(217,169),(218,87),(218,168),(218,170),(219,88),(220,89),(221,106),(221,165),(222,107),(222,166),(223,114),(224,115),(225,78),(225,171),(226,79),(226,172),(227,82),(227,173),(227,229),(228,83),(228,174),(228,230),(229,49),(229,231),(230,50),(230,232),(231,161),(232,162),(233,219),(233,220)],234)
=> ? = 4 - 1
[4,5,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,18),(0,19),(1,17),(1,50),(2,10),(2,13),(3,63),(3,272),(4,284),(5,283),(6,64),(6,287),(7,8),(7,325),(8,309),(9,60),(9,314),(10,38),(10,310),(11,40),(11,149),(12,49),(12,59),(12,321),(13,22),(13,58),(13,310),(14,43),(14,278),(15,41),(15,280),(16,42),(16,313),(17,34),(17,35),(17,304),(18,1),(18,184),(19,12),(19,37),(19,184),(20,231),(20,235),(21,233),(21,298),(22,238),(22,294),(23,68),(23,318),(23,324),(24,123),(24,241),(25,226),(25,237),(26,52),(26,229),(26,322),(27,124),(27,230),(28,127),(28,302),(29,125),(29,303),(30,120),(30,296),(31,232),(31,240),(32,225),(32,293),(33,224),(33,292),(34,243),(34,295),(35,243),(35,316),(36,227),(36,289),(37,51),(37,290),(37,321),(38,242),(38,320),(39,219),(39,300),(40,222),(40,223),(41,220),(41,221),(42,121),(42,297),(43,122),(43,228),(44,66),(44,236),(44,291),(45,65),(45,126),(45,234),(46,67),(46,288),(46,299),(47,54),(47,301),(47,317),(48,55),(48,319),(48,323),(49,53),(49,239),(49,315),(50,242),(50,304),(51,167),(51,308),(52,206),(52,273),(53,255),(53,263),(54,157),(54,253),(55,254),(55,266),(56,181),(56,183),(56,218),(57,182),(57,267),(57,270),(58,238),(58,269),(58,271),(59,239),(59,268),(59,271),(60,106),(60,180),(60,274),(61,89),(61,215),(61,217),(62,90),(62,119),(62,216),(63,145),(63,189),(63,194),(64,57),(64,255),(64,306),(64,316),(65,155),(65,195),(65,197),(66,153),(66,154),(66,156),(67,101),(67,196),(67,262),(68,131),(68,244),(68,245),(69,452),(70,466),(71,417),(72,454),(72,456),(73,451),(73,453),(74,426),(74,449),(75,425),(75,455),(76,414),(76,416),(77,336),(77,456),(78,336),(78,454),(79,335),(79,344),(80,342),(80,450),(81,343),(81,453),(82,11),(83,415),(83,467),(84,413),(84,467),(85,455),(85,465),(86,341),(86,464),(87,472),(88,471),(89,342),(89,441),(90,3),(90,451),(91,357),(92,358),(93,358),(94,394),(94,406),(95,364),(95,452),(96,372),(96,468),(97,371),(97,457),(98,364),(98,424),(99,359),(99,404),(100,376),(100,400),(101,326),(101,377),(102,332),(102,363),(103,331),(103,351),(104,332),(104,374),(105,348),(105,463),(106,341),(106,375),(107,350),(108,409),(109,410),(110,459),(111,459),(112,421),(112,468),(113,458),(114,397),(115,356),(115,408),(116,371),(116,418),(117,372),(117,420),(118,370),(118,411),(119,36),(119,343),(119,451),(120,432),(121,393),(122,369),(123,391),(124,4),(124,392),(125,5),(125,390),(126,46),(126,368),(127,39),(127,365),(128,99),(129,150),(130,149),(131,264),(131,461),(132,178),(132,470),(133,212),(133,462),(134,218),(134,447),(134,449),(135,138),(136,198),(136,436),(137,91),(137,458),(138,92),(138,422),(139,99),(139,407),(139,465),(140,202),(140,464),(141,118),(141,469),(142,97),(142,435),(143,188),(144,100),(144,419),(144,457),(145,200),(145,433),(146,102),(146,345),(146,433),(147,103),(147,346),(147,434),(148,126),(148,344),(149,223),(150,92),(151,193),(152,122),(152,423),(153,163),(153,338),(154,209),(154,337),(155,201),(155,434),(156,240),(156,337),(156,338),(157,265),(158,221),(158,471),(159,93),(160,93),(161,181),(161,444),(161,447),(162,94),(162,405),(162,437),(163,180),(163,438),(164,130),(164,438),(165,175),(165,460),(166,176),(166,337),(166,460),(167,252),(168,106),(168,340),(169,121),(169,339),(170,165),(170,442),(171,210),(171,340),(172,211),(172,339),(173,98),(173,328),(174,79),(174,441),(174,450),(175,78),(175,329),(175,443),(176,77),(176,440),(176,443),(177,69),(177,439),(178,71),(178,328),(179,74),(179,335),(179,432),(180,199),(180,341),(181,288),(181,329),(182,131),(182,327),(182,442),(183,72),(183,329),(183,440),(184,6),(184,290),(185,231),(185,355),(186,203),(186,334),(186,385),(187,105),(187,388),(188,214),(188,382),(189,136),(189,381),(190,213),(190,385),(191,100),(191,389),(192,94),(192,378),(193,108),(193,357),(194,104),(194,381),(194,433),(195,142),(195,384),(196,188),(196,377),(197,205),(197,384),(197,434),(198,222),(198,356),(198,437),(199,186),(199,383),(200,115),(200,367),(201,116),(201,366),(202,109),(202,349),(203,95),(203,380),(203,439),(204,207),(204,382),(205,101),(205,331),(205,395),(206,71),(206,352),(207,83),(207,360),(207,403),(208,70),(208,387),(209,86),(209,362),(210,76),(210,351),(210,401),(211,76),(211,363),(211,402),(212,75),(212,326),(212,349),(213,98),(213,354),(213,380),(214,84),(214,359),(214,360),(215,33),(215,286),(215,342),(216,32),(216,285),(216,343),(217,45),(217,148),(217,441),(218,21),(218,247),(218,440),(219,151),(220,150),(221,159),(222,114),(222,334),(223,177),(223,334),(224,169),(224,446),(225,168),(225,445),(226,143),(227,82),(228,138),(228,369),(229,206),(229,396),(230,107),(230,392),(231,91),(231,373),(232,130),(232,353),(233,132),(233,361),(234,195),(234,368),(235,152),(235,373),(236,31),(236,156),(236,472),(237,26),(237,250),(238,48),(238,249),(238,330),(239,23),(239,248),(239,333),(240,47),(240,276),(240,353),(241,20),(241,185),(241,391),(242,261),(243,7),(243,307),(244,105),(244,256),(244,461),(245,168),(245,171),(246,102),(246,104),(246,466),(247,212),(247,298),(247,347),(248,259),(248,318),(249,259),(249,319),(250,204),(250,322),(251,173),(251,213),(251,470),(252,1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(469,370),(470,141),(470,328),(470,354),(471,159),(471,160),(472,164),(472,232),(472,338),(473,471),(473,481),(474,250),(474,462),(475,458),(475,459),(476,107),(476,478),(477,355),(478,350),(479,88),(479,473),(480,151),(480,469),(481,160),(482,185),(482,477)],483)
=> ? = 6 - 1
[5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? = 2 - 1
[5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,23),(0,24),(1,14),(2,13),(3,35),(3,78),(4,36),(4,79),(5,33),(5,54),(6,34),(6,55),(7,15),(7,80),(8,16),(8,81),(9,17),(9,52),(10,18),(10,53),(11,19),(11,31),(11,82),(12,20),(12,32),(12,83),(13,76),(14,77),(15,58),(16,59),(17,91),(18,92),(19,21),(19,74),(20,22),(20,75),(21,27),(21,88),(22,28),(22,89),(23,11),(23,90),(24,12),(24,90),(25,68),(25,101),(26,69),(26,102),(27,66),(27,97),(28,67),(28,98),(29,47),(29,99),(30,46),(30,100),(31,56),(31,74),(32,57),(32,75),(33,30),(33,97),(33,104),(34,29),(34,98),(34,104),(35,25),(35,84),(35,86),(36,26),(36,85),(36,87),(38,121),(38,122),(39,107),(39,123),(40,108),(40,124),(41,107),(41,108),(42,125),(43,126),(44,116),(44,121),(45,115),(45,122),(46,117),(47,118),(48,125),(49,126),(50,113),(51,114),(52,2),(53,1),(54,3),(55,4),(56,54),(57,55),(58,52),(59,53),(60,50),(60,119),(61,51),(61,120),(62,76),(62,119),(63,77),(63,120),(64,37),(65,37),(66,78),(67,79),(68,62),(68,105),(69,63),(69,106),(70,42),(70,111),(71,43),(71,112),(72,44),(72,110),(73,45),(73,109),(74,7),(74,88),(75,8),(75,89),(76,93),(77,94),(78,84),(79,85),(80,9),(80,58),(81,10),(81,59),(82,5),(82,56),(83,6),(83,57),(84,68),(84,91),(84,123),(85,69),(85,92),(85,124),(86,73),(86,101),(86,123),(87,72),(87,102),(87,124),(88,66),(88,80),(89,67),(89,81),(90,82),(90,83),(91,60),(91,62),(92,61),(92,63),(93,42),(93,48),(94,43),(94,49),(95,39),(95,41),(95,117),(96,40),(96,41),(96,118),(97,46),(97,95),(98,47),(98,96),(99,39),(99,86),(99,118),(100,40),(100,87),(100,117),(101,38),(101,45),(101,105),(102,38),(102,44),(102,106),(103,64),(103,65),(104,95),(104,96),(104,99),(104,100),(105,115),(105,119),(105,121),(106,116),(106,120),(106,122),(107,109),(108,110),(109,50),(109,115),(110,51),(110,116),(111,103),(111,125),(112,103),(112,126),(113,48),(113,111),(114,49),(114,112),(115,113),(115,127),(116,114),(116,127),(117,72),(117,108),(118,73),(118,107),(119,70),(119,93),(119,113),(120,71),(120,94),(120,114),(121,70),(121,127),(122,71),(122,127),(123,60),(123,105),(123,109),(124,61),(124,106),(124,110),(125,64),(126,65),(127,111),(127,112)],128)
=> ? = 4 - 1
[5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,18),(0,19),(1,12),(1,17),(2,7),(3,105),(4,11),(4,167),(5,153),(6,145),(7,104),(8,41),(8,154),(9,10),(9,173),(10,166),(11,136),(12,80),(13,6),(13,168),(14,26),(14,102),(15,23),(15,162),(16,27),(16,28),(16,171),(17,22),(17,25),(17,80),(18,1),(18,103),(19,16),(19,33),(19,103),(20,131),(20,160),(21,66),(21,159),(22,134),(22,170),(23,127),(23,156),(24,126),(24,133),(25,125),(25,134),(26,128),(26,129),(27,70),(27,158),(28,37),(28,70),(28,169),(29,67),(29,68),(30,42),(30,157),(30,161),(31,36),(31,130),(31,172),(32,34),(32,69),(32,132),(33,35),(33,155),(33,171),(34,112),(34,113),(35,85),(35,165),(36,119),(36,148),(37,137),(37,141),(38,98),(38,101),(38,124),(39,100),(39,143),(39,144),(40,58),(40,99),(40,123),(41,39),(41,137),(41,163),(41,170),(42,62),(42,111),(42,140),(44,239),(45,229),(45,230),(46,214),(47,213),(48,182),(48,228),(49,179),(49,230),(50,179),(50,229),(51,177),(51,183),(52,178),(53,216),(53,240),(54,218),(54,233),(55,217),(56,215),(56,240),(57,242),(58,182),(58,225),(59,189),(59,209),(60,186),(60,236),(61,190),(61,207),(62,181),(62,197),(63,190),(63,237),(64,211),(64,212),(65,210),(66,220),(67,196),(68,5),(68,196),(69,30),(69,195),(70,15),(70,226),(71,114),(71,239),(72,121),(73,124),(73,232),(73,233),(74,98),(74,231),(74,232),(75,116),(75,241),(76,95),(76,238),(77,107),(77,236),(78,59),(79,69),(79,183),(80,14),(80,125),(81,109),(82,104),(82,242),(83,93),(83,235),(84,94),(84,180),(84,235),(85,139),(86,59),(86,208),(86,237),(87,83),(87,222),(88,52),(88,221),(89,55),(89,178),(90,43),(91,43),(92,51),(92,225),(92,228),(93,50),(93,175),(93,223),(94,49),(94,223),(94,224),(95,47),(95,176),(96,54),(96,177),(96,220),(97,46),(97,176),(98,157),(98,175),(99,108),(99,182),(100,127),(100,174),(100,222),(101,45),(101,175),(101,224),(102,3),(102,129),(103,8),(103,155),(104,90),(105,4),(105,150),(106,117),(106,198),(107,62),(107,203),(108,147),(108,201),(109,122),(109,198),(110,64),(110,191),(111,109),(111,197),(112,88),(112,199),(113,107),(113,199),(114,65),(114,194),(115,60),(115,202),(116,61),(116,181),(116,194),(117,53),(117,188),(117,206),(118,55),(118,192),(119,47),(119,185),(120,44),(120,187),(121,46),(121,200),(122,56),(122,188),(122,189),(123,32),(123,79),(123,225),(124,20),(124,138),(124,224),(125,102),(126,81),(127,142),(127,234),(128,120),(128,180),(129,105),(129,180),(130,119),(130,204),(131,76),(131,193),(132,112),(132,195),(133,31),(133,146),(134,9),(134,164),(135,97),(135,121),(135,238),(136,95),(136,97),(137,87),(137,100),(137,227),(138,116),(138,160),(138,184),(139,66),(139,96),(140,63),(140,86),(140,181),(141,51),(141,79),(141,227),(142,44),(142,71),(143,84),(143,128),(143,222),(144,54),(144,73),(144,174),(145,52),(145,89),(146,106),(146,172),(147,77),(147,113),(147,202),(148,53),(148,56),(148,185),(149,78),(149,86),(150,152),(150,167),(151,89),(151,118),(151,221),(152,72),(152,135),(153,57),(153,82),(154,21),(154,139),(154,163),(155,85),(155,154),(156,60),(156,77),(156,234),(157,140),(157,149),(158,48),(158,99),(158,226),(159,38),(159,73),(159,74),(159,220),(160,61),(160,63),(160,193),(161,81),(161,111),(162,115),(162,147),(162,156),(163,96),(163,144),(163,159),(163,227),(164,83),(164,84),(164,173),(165,48),(165,58),(165,92),(166,45),(166,49),(166,50),(167,76),(167,135),(167,136),(168,88),(168,145),(168,151),(169,92),(169,123),(169,141),(169,226),(170,87),(170,143),(170,164),(171,40),(171,158),(171,165),(171,169),(172,117),(172,122),(172,148),(172,204),(173,93),(173,94),(173,101),(173,166),(174,218),(174,232),(174,234),(175,149),(175,229),(176,64),(176,213),(176,214),(177,218),(177,231),(178,67),(178,217),(179,65),(179,205),(180,150),(180,187),(181,190),(181,208),(182,13),(182,201),(183,195),(184,193),(184,194),(184,205),(185,213),(185,215),(185,216),(186,241),(187,152),(187,239),(188,153),(188,240),(188,246),(189,215),(189,246),(190,245),(191,211),(192,68),(192,217),(193,207),(193,237),(193,238),(194,207),(194,208),(194,210),(195,161),(196,2),(197,198),(198,188),(199,203),(199,221),(200,191),(200,214),(201,168),(201,202),(202,151),(202,199),(202,236),(203,192),(203,197),(204,185),(204,189),(204,206),(205,204),(205,210),(206,216),(206,246),(207,200),(207,245),(208,209),(208,245),(209,246),(210,206),(210,209),(211,247),(212,247),(213,212),(213,219),(214,211),(214,219),(215,212),(215,244),(216,219),(216,244),(217,196),(218,186),(218,243),(219,247),(220,24),(220,231),(220,233),(221,29),(221,178),(221,192),(222,120),(222,142),(222,235),(223,114),(223,179),(223,184),(224,131),(224,184),(224,230),(225,132),(225,183),(226,108),(226,162),(226,228),(227,74),(227,174),(227,177),(228,115),(228,201),(229,78),(230,130),(230,205),(231,126),(231,243),(232,75),(232,138),(232,243),(233,133),(233,243),(234,75),(234,186),(235,71),(235,187),(235,223),(236,118),(236,203),(237,110),(237,245),(238,110),(238,176),(238,200),(239,72),(240,82),(240,244),(241,106),(242,90),(242,91),(243,146),(243,241),(244,242),(244,247),(245,191),(246,57),(246,244),(247,91)],248)
=> ? = 4 - 1
[5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,25),(0,26),(0,66),(1,56),(2,55),(3,7),(3,200),(4,168),(5,303),(6,260),(7,264),(8,70),(8,306),(9,10),(9,68),(9,201),(10,18),(10,307),(11,48),(11,203),(12,47),(12,337),(13,52),(13,304),(14,53),(14,289),(15,46),(15,240),(16,44),(16,98),(17,54),(17,202),(18,45),(18,335),(19,51),(19,336),(20,24),(20,50),(20,204),(21,49),(21,344),(22,36),(22,338),(23,37),(23,38),(23,341),(24,34),(24,43),(24,253),(25,9),(25,276),(26,23),(26,40),(26,163),(27,256),(27,325),(28,130),(28,131),(29,76),(29,258),(29,327),(30,127),(30,323),(31,62),(31,129),(31,324),(32,75),(32,319),(32,328),(33,310),(33,326),(34,259),(34,342),(35,77),(35,245),(35,318),(36,242),(36,314),(37,133),(37,322),(38,64),(38,133),(38,339),(39,74),(39,254),(39,343),(40,73),(40,315),(40,341),(41,60),(41,257),(41,320),(42,59),(42,132),(42,329),(43,57),(43,259),(43,312),(44,63),(44,316),(44,345),(45,321),(45,330),(46,65),(46,330),(46,345),(47,251),(47,309),(48,246),(48,252),(49,241),(49,340),(50,249),(50,253),(51,250),(51,313),(52,244),(52,247),(53,126),(53,248),(54,61),(54,311),(54,317),(55,128),(55,255),(56,58),(56,243),(56,308),(57,162),(57,316),(58,216),(58,293),(59,214),(59,215),(60,159),(60,161),(61,160),(61,286),(62,140),(62,261),(63,155),(63,273),(64,274),(64,287),(65,96),(65,282),(66,20),(66,163),(66,276),(67,196),(67,198),(67,239),(68,197),(68,199),(68,307),(69,233),(69,234),(69,288),(70,72),(70,287),(70,334),(70,342),(71,125),(71,195),(71,238),(72,157),(72,267),(72,280),(73,173),(73,199),(73,333),(74,213),(74,236),(74,295),(75,102),(75,210),(75,262),(76,94),(76,150),(76,277),(77,69),(77,279),(77,291),(77,294),(78,432),(79,463),(80,465),(81,430),(82,366),(82,466),(83,367),(84,442),(84,464),(85,434),(85,481),(86,431),(87,480),(88,441),(89,438),(89,470),(90,429),(90,433),(91,364),(91,473),(92,354),(92,467),(93,354),(93,462),(94,353),(94,365),(95,469),(95,470),(96,468),(97,487),(98,17),(99,359),(99,385),(100,382),(101,387),(101,425),(102,362),(102,404),(103,394),(103,436),(104,410),(104,435),(105,384),(105,419),(106,376),(106,423),(107,380),(107,474),(108,364),(108,389),(109,394),(109,482),(110,384),(110,476),(111,356),(111,401),(112,389),(112,453),(113,14),(113,391),(114,375),(114,387),(115,351),(116,427),(116,428),(117,426),(118,476),(119,488),(120,403),(121,381),(122,391),(123,385),(123,402),(124,388),(125,33),(125,357),(125,366),(126,352),(127,6),(127,446),(128,400),(129,13),(129,373),(130,5),(130,392),(131,4),(131,392),(132,32),(132,409),(133,19),(133,457),(134,191),(134,352),(135,153),(135,479),(136,218),(136,480),(137,232),(138,239),(138,459),(138,464),(139,171),(139,460),(140,175),(140,485),(141,128),(142,123),(142,478),(143,121),(143,463),(144,114),(144,439),(144,481),(145,100),(145,465),(146,99),(146,486),(147,237),(148,222),(148,475),(149,158),(149,447),(150,180),(150,353),(151,104),(151,351),(151,365),(152,111),(152,357),(152,466),(153,105),(153,350),(154,209),(154,474),(155,241),(155,363),(156,243),(157,250),(157,347),(157,460),(158,126),(158,349),(159,166),(159,360),(160,176),(160,346),(161,129),(161,360),(162,202),(163,8),(163,315),(164,255),(164,487),(165,106),(165,452),(166,242),(166,449),(167,318),(168,141),(169,101),(169,418),(170,134),(171,189),(171,472),(172,145),(172,450),(173,272),(173,451),(174,270),(174,449),(175,156),(176,251),(176,458),(177,120),(177,458),(178,99),(178,443),(178,483),(179,106),(179,437),(179,482),(180,308),(180,452),(181,142),(181,448),(182,190),(182,358),(182,472),(183,94),(183,462),(183,467),(184,86),(185,86),(185,440),(186,88),(186,349),(187,78),(188,79),(188,355),(189,93),(189,348),(189,455),(190,92),(190,454),(190,455),(191,80),(191,450),(192,84),(192,356),(192,446),(193,81),(193,350),(194,83),(194,456),(195,217),(195,366),(196,319),(196,348),(197,240),(197,451),(198,183),(198,348),(198,454),(199,266),(199,451),(200,12),(200,297),(201,15),(201,197),(202,317),(203,3),(203,246),(204,16),(204,249),(205,228),(205,396),(206,107),(206,408),(207,196),(207,390),(207,459),(208,110),(208,397),(209,102),(209,415),(210,221),(210,404),(211,104),(211,393),(211,469),(212,116),(212,395),(213,205),(213,414),(214,149),(214,406),(215,209),(215,406),(216,100),(216,412),(217,290),(217,407),(218,117),(218,371),(219,220),(219,379),(220,213),(220,410),(220,483),(221,235),(221,398),(222,103),(222,362),(222,371),(223,227),(223,398),(224,132),(224,401),(225,244),(225,378),(225,448),(226,87),(226,369),(227,85),(227,377),(227,424),(228,90),(228,375),(228,420),(229,88),(229,386),(230,89),(230,374),(231,78),(231,370),(232,81),(232,399),(233,225),(233,361),(233,413),(234,219),(234,413),(235,144),(235,376),(235,377),(236,105),(236,383),(236,414),(237,90),(237,381),(237,382),(238,42),(238,224),(238,357),(239,27),(239,268),(239,454),(240,35),(240,167),(241,300),(242,170),(243,216),(243,359),(244,284),(244,477),(245,294),(245,367),(246,200),(246,358),(247,285),(247,477),(248,172),(248,352),(249,98),(250,283),(250,471),(251,265),(252,226),(252,358),(253,11),(253,312),(254,236),(254,416),(255,231),(255,400),(256,135),(256,372),(257,31),(257,161),(257,456),(258,39),(258,269),(259,21),(259,332),(260,119),(260,278),(261,181),(261,225),(261,485),(262,109),(262,179),(262,362),(263,193),(263,232),(263,479),(264,176),(264,177),(265,110),(265,118),(266,194),(266,320),(267,182),(267,252),(267,460),(268,222),(268,325),(268,368),(269,223),(269,343),(270,175),(270,289),(271,180),(271,221),(272,127),(272,192),(273,112),(273,279),(273,363),(274,111),(274,224),(274,461),(275,108),(275,112),(275,468),(276,201),(276,204),(277,178),(277,220),(277,365),(278,219),(278,277),(278,488),(279,288),(279,298),(279,453),(280,84),(280,138),(280,347),(281,91),(281,108),(281,363),(282,91),(282,311),(282,468),(283,87),(283,136),(284,79),(284,143),(285,80),(285,145),(286,89),(286,95),(286,346),(287,139),(287,157),(287,461),(288,95),(288,211),(288,361),(289,156),(289,248),(290,154),(290,215),(290,408),(291,234),(291,278),(291,405),(292,228),(292,237),(292,411),(293,101),(293,114),(293,412),(294,233),(294,261),(294,405),(294,453),(295,85),(295,144),(295,383),(296,165),(296,179),(297,302),(297,337),(298,151),(298,211),(298,484),(299,186),(299,229),(299,447),(300,115),(300,151),(301,134),(301,331),(302,137),(302,263),(303,97),(303,164),(304,247),(304,331),(305,184),(305,185),(306,30),(306,272),(306,334),(307,41),(307,266),(307,335),(308,169),(308,293),(308,359),(309,153),(309,193),(310,166),(310,174),(311,230),(311,286),(311,473),(312,162),(312,203),(313,107),(313,154),(313,471),(314,158),(314,186),(315,173),(315,306),(316,155),(316,281),(317,160),(317,264),(317,473),(318,260),(318,291),(318,367),(319,262),(319,296),(320,159),(320,310),(320,456),(321,83),(321,245),(322,82),(322,195),(322,457),(323,67),(323,138),(323,207),(323,446),(324,140),(324,270),(324,373),(325,103),(325,109),(325,372),(326,113),(326,122),(327,150),(327,271),(328,210),(328,271),(329,113),(329,214),(329,409),(330,96),(330,275),(331,172),(331,191),(331,285),(332,171),(332,182),(332,344),(333,82),(333,125),(333,152),(334,192),(334,280),(334,323),(334,461),(335,194),(335,257),(335,321),(336,206),(336,290),(336,313),(337,135),(337,263),(337,309),(338,149),(338,299),(338,314),(339,152),(339,238),(339,274),(339,457),(340,92),(340,93),(340,183),(341,71),(341,322),(341,333),(341,339),(342,139),(342,267),(342,332),(343,227),(343,235),(343,295),(343,416),(344,189),(344,190),(344,198),(344,340),(345,273),(345,275),(345,281),(345,282),(346,438),(346,458),(346,469),(347,442),(347,459),(347,471),(348,296),(348,462),(349,131),(349,441),(350,116),(350,419),(350,430),(351,123),(351,435),(351,443),(352,450),(353,452),(354,117),(354,422),(355,305),(355,463),(356,390),(356,442),(357,326),(357,329),(357,401),(358,297),(358,369),(359,412),(359,418),(360,373),(360,449),(361,393),(361,448),(361,470),(362,394),(362,437),(363,298),(363,300),(363,389),(364,374),(365,410),(365,443),(366,22),(366,407),(367,119),(367,405),(368,371),(368,372),(368,422),(369,302),(369,480),(370,432),(371,426),(371,436),(371,437),(372,436),(372,479),(372,482),(373,301),(373,304),(374,120),(374,438),(375,429),(375,496),(376,439),(376,495),(377,303),(377,481),(377,495),(378,477),(378,486),(379,483),(379,486),(380,475),(381,433),(381,444),(382,429),(382,444),(383,419),(383,434),(383,439),(384,417),(385,418),(386,130),(386,441),(387,124),(387,496),(388,421),(389,115),(389,484),(390,327),(390,491),(391,1),(392,2),(393,435),(393,478),(393,494),(394,493),(395,427),(396,417),(396,420),(397,305),(397,476),(398,377),(399,395),(399,430),(400,370),(401,122),(401,409),(402,121),(402,411),(403,118),(403,397),(404,398),(405,413),(405,485),(405,488),(406,415),(406,447),(407,338),(407,408),(408,299),(408,406),(408,474),(409,328),(409,391),(410,414),(410,492),(411,381),(411,420),(412,375),(412,382),(412,425),(413,378),(413,379),(414,384),(414,396),(415,386),(415,404),(416,376),(416,383),(416,424),(417,440),(418,425),(419,428),(419,445),(420,433),(420,440),(420,496),(421,432),(422,416),(422,426),(423,495),(424,434),(424,495),(425,444),(425,496),(426,423),(426,424),(427,497),(428,124),(428,497),(429,400),(429,490),(430,427),(430,445),(431,421),(433,431),(433,490),(434,445),(434,489),(435,402),(435,492),(436,399),(436,493),(437,423),(437,493),(438,403),(438,494),(439,387),(439,428),(439,489),(440,388),(440,431),(441,392),(442,380),(442,491),(443,385),(443,492),(444,490),(445,497),(446,29),(446,390),(446,464),(447,28),(447,349),(447,386),(448,188),(448,284),(448,478),(449,170),(449,301),(450,168),(450,465),(451,167),(452,169),(453,181),(453,361),(453,484),(454,256),(454,368),(454,467),(455,218),(455,354),(455,368),(456,174),(456,324),(456,360),(457,217),(457,336),(457,466),(458,208),(458,265),(458,403),(459,148),(459,268),(459,491),(460,226),(460,283),(460,472),(461,207),(461,347),(461,356),(462,165),(462,353),(463,184),(464,258),(464,491),(465,141),(466,206),(466,407),(467,254),(467,422),(468,230),(468,364),(469,208),(469,494),(470,188),(470,494),(471,148),(471,380),(472,136),(472,369),(472,455),(473,177),(473,346),(473,374),(474,229),(474,415),(475,223),(476,185),(476,417),(477,147),(478,143),(478,355),(478,402),(479,212),(479,350),(479,399),(480,137),(481,164),(481,489),(482,212),(482,493),(483,205),(483,292),(483,492),(484,142),(484,351),(484,393),(485,146),(485,378),(486,147),(486,292),(487,187),(487,231),(488,146),(488,178),(488,379),(489,487),(489,497),(490,370),(490,421),(491,269),(491,475),(492,396),(492,411),(493,395),(494,355),(494,397),(495,97),(495,489),(496,388),(496,490),(497,187)],498)
=> ? = 6 - 1
[5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? = 4 - 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
Description
The number of posets with combinatorially isomorphic order polytopes.
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