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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000047
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St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 3
[2,2] => 3
[3,1] => 3
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 3
[1,2,2] => 3
[1,3,1] => 3
[1,4] => 1
[2,1,1,1] => 4
[2,1,2] => 4
[2,2,1] => 8
[2,3] => 4
[3,1,1] => 6
[3,2] => 6
[4,1] => 4
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 1
[1,1,2,1,1] => 3
[1,1,2,2] => 3
[1,1,3,1] => 3
[1,1,4] => 1
[1,2,1,1,1] => 4
[1,2,1,2] => 4
[1,2,2,1] => 8
[1,2,3] => 4
[1,3,1,1] => 6
[1,3,2] => 6
[1,4,1] => 4
[1,5] => 1
[2,1,1,1,1] => 5
[2,1,1,2] => 5
[2,1,2,1] => 10
Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000100
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 38%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
=> [.,.]
=> ([],1)
=> ? = 1
[1,1] => [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 8
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 6
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 6
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 8
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 10
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 3
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 3
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 3
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 4
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 4
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ? = 8
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 4
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ? = 6
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ? = 6
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 4
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 5
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 5
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ? = 10
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 5
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 15
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 15
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 15
[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 5
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 10
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 10
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(6,7),(7,4)],8)
=> ? = 20
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 10
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 10
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 10
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 5
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 6
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 6
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,7),(4,5),(5,7),(6,4),(7,3)],8)
=> ? = 12
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 6
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ? = 18
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ? = 18
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ? = 18
[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 6
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ? = 24
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[[.,.],[.,[[.,.],.]]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ? = 24
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ? = 48
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ? = 24
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> ([(0,6),(1,4),(2,3),(3,7),(4,7),(6,5),(7,6)],8)
=> ? = 36
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> ([(0,6),(1,4),(2,3),(3,7),(4,7),(6,5),(7,6)],8)
=> ? = 36
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ? = 24
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 6
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ? = 15
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ? = 15
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(5,7),(6,5),(7,4)],8)
=> ? = 30
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ? = 15
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ? = 45
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ? = 45
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ? = 45
Description
The number of linear extensions of a poset.
Matching statistic: St000255
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000255: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 23%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000255: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 23%
Values
[1] => [1] => [1,0]
=> [1] => 1
[1,1] => [1,1] => [1,0,1,0]
=> [1,2] => 1
[2] => [2] => [1,1,0,0]
=> [2,1] => 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[3] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 8
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 4
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 6
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 6
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 4
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 3
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 3
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 3
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 4
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 4
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 8
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => 4
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => 6
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => 6
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 4
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 5
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 5
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 10
[1,1,1,2,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ? = 3
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,1,1,3,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7] => ? = 3
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ? = 4
[1,1,2,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 4
[1,1,2,2,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => ? = 8
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 4
[1,1,3,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ? = 6
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => ? = 6
[1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 4
[1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 5
[1,2,1,2,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => ? = 10
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 5
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => ? = 15
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 15
[1,2,3,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,3,6,5,7] => ? = 15
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 5
[1,3,1,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ? = 10
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 10
[1,3,2,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => ? = 20
[1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 10
[1,4,1,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ? = 10
[1,5,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 5
[2,1,1,1,2] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 6
[2,1,1,2,1] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => ? = 12
[2,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ? = 18
[2,1,2,2] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 18
[2,1,3,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => ? = 18
[2,2,1,1,1] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 24
[2,2,1,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 24
[2,2,2,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 48
[2,2,3] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 24
[2,3,1,1] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => ? = 36
[2,3,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 36
[3,1,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => ? = 15
[3,1,2,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => ? = 30
[3,1,3] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 15
[3,2,1,1] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => ? = 45
[3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 45
[3,3,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 45
[3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 15
[4,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 20
[4,2,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 40
[4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 20
[5,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 15
[7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ? = 1
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => ? = 2
[1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 1
Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Matching statistic: St000045
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000045: Binary trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 21%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000045: Binary trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 21%
Values
[1] => [1,0]
=> [.,.]
=> ? = 1
[1,1] => [1,0,1,0]
=> [.,[.,.]]
=> ? = 1
[2] => [1,1,0,0]
=> [[.,.],.]
=> ? = 1
[1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 8
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 6
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 6
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 8
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 10
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 5
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 15
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 15
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 3
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 3
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 3
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 4
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 4
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 8
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 4
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 6
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 6
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 4
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ? = 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 5
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ? = 5
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 10
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,[[[.,.],.],.]]]]
=> ? = 5
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 15
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 15
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> ? = 15
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ? = 5
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ? = 10
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [.,[[[.,.],.],[.,[[.,.],.]]]]
=> ? = 10
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> ? = 20
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],[[[.,.],.],.]]]
=> ? = 10
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> ? = 10
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [.,[[[[.,.],.],.],[[.,.],.]]]
=> ? = 10
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> ? = 5
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 6
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 6
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 12
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,[[[.,.],.],.]]]]
=> ? = 6
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ? = 18
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],[[.,.],.]]]]
=> ? = 18
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[.,.],[.,[[[.,.],.],[.,.]]]]
=> ? = 18
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[[[[.,.],.],.],.]]]
=> ? = 6
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 24
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[[.,.],[.,[[.,.],.]]]]
=> ? = 24
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[[.,.],[.,.]]]]
=> ? = 48
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[.,.],[[[.,.],.],.]]]
=> ? = 24
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ? = 36
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[[[.,.],.],[[.,.],.]]]
=> ? = 36
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[.,.],[[[[.,.],.],.],[.,.]]]
=> ? = 24
Description
The number of linear extensions of a binary tree.
Also, the number of increasing / decreasing binary trees labelled by $1, \dots, n$ of this shape.
Also, the size of the sylvester class corresponding to this tree when the Tamari order is seen as a quotient poset of the right weak order on permutations.
Also, the number of permutations which give this tree shape when inserted in a binary search tree.
Also, the number of permutations which increasing / decreasing tree is of this shape.
Matching statistic: St001881
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 6%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 6%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,1] => [2] => ([],2)
=> ([],1)
=> 1
[2] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1] => [3] => ([],3)
=> ([],1)
=> 1
[1,2] => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,1] => [4] => ([],4)
=> ([],1)
=> 1
[1,1,2] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 1
[1,1,1,1,1] => [5] => ([],5)
=> ([],1)
=> 1
[1,1,1,2] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3
[1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 4
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 4
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 6
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 6
[4,1] => [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),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 4
[5] => [1,1,1,1,1] => ([(0,1),(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),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? = 1
[1,1,1,1,1,1] => [6] => ([],6)
=> ([],1)
=> 1
[1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3
[1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 4
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 4
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 6
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 6
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 4
[1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? = 1
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 5
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,29),(2,13),(2,14),(2,15),(2,29),(3,10),(3,11),(3,12),(3,29),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,19),(7,22),(7,25),(7,28),(8,19),(8,20),(8,23),(8,26),(9,19),(9,21),(9,24),(9,27),(10,20),(10,22),(10,30),(11,21),(11,22),(11,31),(12,20),(12,21),(12,32),(13,23),(13,25),(13,30),(14,24),(14,25),(14,31),(15,23),(15,24),(15,32),(16,26),(16,28),(16,30),(17,27),(17,28),(17,31),(18,26),(18,27),(18,32),(19,33),(19,34),(19,35),(20,33),(20,36),(21,33),(21,37),(22,33),(22,38),(23,34),(23,36),(24,34),(24,37),(25,34),(25,38),(26,35),(26,36),(27,35),(27,37),(28,35),(28,38),(29,30),(29,31),(29,32),(30,36),(30,38),(31,37),(31,38),(32,36),(32,37),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 5
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(1,9),(1,36),(1,37),(2,12),(2,16),(2,21),(2,22),(2,37),(3,11),(3,15),(3,19),(3,20),(3,37),(4,14),(4,18),(4,20),(4,22),(4,36),(5,13),(5,17),(5,19),(5,21),(5,36),(6,9),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(8,35),(8,38),(8,39),(9,35),(9,40),(9,41),(10,31),(10,32),(10,33),(10,34),(10,35),(11,23),(11,24),(11,31),(11,38),(12,25),(12,26),(12,32),(12,38),(13,23),(13,25),(13,33),(13,39),(14,24),(14,26),(14,34),(14,39),(15,27),(15,28),(15,31),(15,40),(16,29),(16,30),(16,32),(16,40),(17,27),(17,29),(17,33),(17,41),(18,28),(18,30),(18,34),(18,41),(19,23),(19,27),(19,48),(20,24),(20,28),(20,48),(21,25),(21,29),(21,48),(22,26),(22,30),(22,48),(23,42),(23,49),(24,43),(24,49),(25,44),(25,49),(26,45),(26,49),(27,42),(27,50),(28,43),(28,50),(29,44),(29,50),(30,45),(30,50),(31,42),(31,43),(31,46),(32,44),(32,45),(32,46),(33,42),(33,44),(33,47),(34,43),(34,45),(34,47),(35,46),(35,47),(36,39),(36,41),(36,48),(37,38),(37,40),(37,48),(38,46),(38,49),(39,47),(39,49),(40,46),(40,50),(41,47),(41,50),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,49),(48,50),(49,51),(50,51)],52)
=> ? = 10
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(1,17),(1,24),(1,38),(1,40),(2,10),(2,16),(2,24),(2,37),(2,39),(3,12),(3,18),(3,23),(3,37),(3,40),(4,13),(4,19),(4,23),(4,38),(4,39),(5,15),(5,21),(5,22),(5,39),(5,40),(6,14),(6,20),(6,22),(6,37),(6,38),(7,9),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,31),(9,32),(9,33),(9,34),(9,35),(9,36),(10,25),(10,31),(10,41),(10,43),(11,25),(11,32),(11,42),(11,44),(12,26),(12,33),(12,41),(12,44),(13,26),(13,34),(13,42),(13,43),(14,27),(14,35),(14,41),(14,42),(15,27),(15,36),(15,43),(15,44),(16,28),(16,31),(16,45),(16,47),(17,28),(17,32),(17,46),(17,48),(18,29),(18,33),(18,45),(18,48),(19,29),(19,34),(19,46),(19,47),(20,30),(20,35),(20,45),(20,46),(21,30),(21,36),(21,47),(21,48),(22,27),(22,30),(22,56),(23,26),(23,29),(23,56),(24,25),(24,28),(24,56),(25,49),(25,57),(26,50),(26,57),(27,51),(27,57),(28,49),(28,58),(29,50),(29,58),(30,51),(30,58),(31,49),(31,52),(31,54),(32,49),(32,53),(32,55),(33,50),(33,52),(33,55),(34,50),(34,53),(34,54),(35,51),(35,52),(35,53),(36,51),(36,54),(36,55),(37,41),(37,45),(37,56),(38,42),(38,46),(38,56),(39,43),(39,47),(39,56),(40,44),(40,48),(40,56),(41,52),(41,57),(42,53),(42,57),(43,54),(43,57),(44,55),(44,57),(45,52),(45,58),(46,53),(46,58),(47,54),(47,58),(48,55),(48,58),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,59),(56,57),(56,58),(57,59),(58,59)],60)
=> ? = 5
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,11),(2,15),(2,20),(2,21),(2,23),(2,50),(3,10),(3,14),(3,18),(3,19),(3,22),(3,50),(4,13),(4,17),(4,19),(4,21),(4,25),(4,49),(5,12),(5,16),(5,18),(5,20),(5,24),(5,49),(6,14),(6,15),(6,16),(6,17),(6,27),(6,48),(7,10),(7,11),(7,12),(7,13),(7,26),(7,48),(8,9),(8,48),(8,49),(8,50),(9,59),(9,60),(9,61),(10,28),(10,40),(10,41),(10,63),(11,29),(11,42),(11,43),(11,63),(12,30),(12,40),(12,42),(12,64),(13,31),(13,41),(13,43),(13,64),(14,32),(14,44),(14,45),(14,63),(15,33),(15,46),(15,47),(15,63),(16,34),(16,44),(16,46),(16,64),(17,35),(17,45),(17,47),(17,64),(18,36),(18,40),(18,44),(18,62),(19,37),(19,41),(19,45),(19,62),(20,38),(20,42),(20,46),(20,62),(21,39),(21,43),(21,47),(21,62),(22,28),(22,32),(22,36),(22,37),(22,59),(23,29),(23,33),(23,38),(23,39),(23,59),(24,30),(24,34),(24,36),(24,38),(24,60),(25,31),(25,35),(25,37),(25,39),(25,60),(26,28),(26,29),(26,30),(26,31),(26,61),(27,32),(27,33),(27,34),(27,35),(27,61),(28,51),(28,52),(28,66),(29,53),(29,54),(29,66),(30,51),(30,53),(30,67),(31,52),(31,54),(31,67),(32,55),(32,56),(32,66),(33,57),(33,58),(33,66),(34,55),(34,57),(34,67),(35,56),(35,58),(35,67),(36,51),(36,55),(36,65),(37,52),(37,56),(37,65),(38,53),(38,57),(38,65),(39,54),(39,58),(39,65),(40,51),(40,68),(41,52),(41,68),(42,53),(42,68),(43,54),(43,68),(44,55),(44,68),(45,56),(45,68),(46,57),(46,68),(47,58),(47,68),(48,61),(48,63),(48,64),(49,60),(49,62),(49,64),(50,59),(50,62),(50,63),(51,69),(52,69),(53,69),(54,69),(55,69),(56,69),(57,69),(58,69),(59,65),(59,66),(60,65),(60,67),(61,66),(61,67),(62,65),(62,68),(63,66),(63,68),(64,67),(64,68),(65,69),(66,69),(67,69),(68,69)],70)
=> ? = 15
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
=> ? = 15
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
=> ? = 15
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,13),(1,24),(1,25),(1,31),(1,54),(1,55),(1,57),(2,12),(2,22),(2,23),(2,30),(2,52),(2,53),(2,57),(3,15),(3,27),(3,29),(3,33),(3,53),(3,55),(3,56),(4,14),(4,26),(4,28),(4,32),(4,52),(4,54),(4,56),(5,16),(5,22),(5,26),(5,35),(5,55),(5,58),(5,60),(6,17),(6,23),(6,27),(6,36),(6,54),(6,58),(6,61),(7,18),(7,24),(7,28),(7,36),(7,53),(7,59),(7,60),(8,19),(8,25),(8,29),(8,35),(8,52),(8,59),(8,61),(9,21),(9,32),(9,33),(9,34),(9,57),(9,60),(9,61),(10,20),(10,30),(10,31),(10,34),(10,56),(10,58),(10,59),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(12,37),(12,38),(12,45),(12,72),(12,73),(12,77),(13,39),(13,40),(13,46),(13,74),(13,75),(13,77),(14,41),(14,43),(14,47),(14,72),(14,74),(14,76),(15,42),(15,44),(15,48),(15,73),(15,75),(15,76),(16,37),(16,41),(16,49),(16,75),(16,78),(16,80),(17,38),(17,42),(17,50),(17,74),(17,78),(17,81),(18,39),(18,43),(18,50),(18,73),(18,79),(18,80),(19,40),(19,44),(19,49),(19,72),(19,79),(19,81),(20,45),(20,46),(20,51),(20,76),(20,78),(20,79),(21,47),(21,48),(21,51),(21,77),(21,80),(21,81),(22,37),(22,62),(22,66),(22,96),(23,38),(23,62),(23,67),(23,95),(24,39),(24,63),(24,68),(24,96),(25,40),(25,63),(25,69),(25,95),(26,41),(26,64),(26,66),(26,94),(27,42),(27,65),(27,67),(27,94),(28,43),(28,64),(28,68),(28,93),(29,44),(29,65),(29,69),(29,93),(30,45),(30,62),(30,70),(30,93),(31,46),(31,63),(31,70),(31,94),(32,47),(32,64),(32,71),(32,95),(33,48),(33,65),(33,71),(33,96),(34,51),(34,70),(34,71),(34,92),(35,49),(35,66),(35,69),(35,92),(36,50),(36,67),(36,68),(36,92),(37,82),(37,86),(37,100),(38,82),(38,87),(38,99),(39,83),(39,88),(39,100),(40,83),(40,89),(40,99),(41,84),(41,86),(41,98),(42,85),(42,87),(42,98),(43,84),(43,88),(43,97),(44,85),(44,89),(44,97),(45,82),(45,90),(45,97),(46,83),(46,90),(46,98),(47,84),(47,91),(47,99),(48,85),(48,91),(48,100),(49,86),(49,89),(49,101),(50,87),(50,88),(50,101),(51,90),(51,91),(51,101),(52,66),(52,72),(52,93),(52,95),(53,67),(53,73),(53,93),(53,96),(54,68),(54,74),(54,94),(54,95),(55,69),(55,75),(55,94),(55,96),(56,71),(56,76),(56,93),(56,94),(57,70),(57,77),(57,95),(57,96),(58,62),(58,78),(58,92),(58,94),(59,63),(59,79),(59,92),(59,93),(60,64),(60,80),(60,92),(60,96),(61,65),(61,81),(61,92),(61,95),(62,82),(62,102),(63,83),(63,102),(64,84),(64,102),(65,85),(65,102),(66,86),(66,102),(67,87),(67,102),(68,88),(68,102),(69,89),(69,102),(70,90),(70,102),(71,91),(71,102),(72,86),(72,97),(72,99),(73,87),(73,97),(73,100),(74,88),(74,98),(74,99),(75,89),(75,98),(75,100),(76,91),(76,97),(76,98),(77,90),(77,99),(77,100),(78,82),(78,98),(78,101),(79,83),(79,97),(79,101),(80,84),(80,100),(80,101),(81,85),(81,99),(81,101),(82,103),(83,103),(84,103),(85,103),(86,103),(87,103),(88,103),(89,103),(90,103),(91,103),(92,101),(92,102),(93,97),(93,102),(94,98),(94,102),(95,99),(95,102),(96,100),(96,102),(97,103),(98,103),(99,103),(100,103),(101,103),(102,103)],104)
=> ? = 5
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,15),(1,20),(1,21),(1,23),(1,29),(1,69),(2,10),(2,14),(2,18),(2,19),(2,22),(2,28),(2,69),(3,13),(3,17),(3,19),(3,21),(3,25),(3,31),(3,68),(4,12),(4,16),(4,18),(4,20),(4,24),(4,30),(4,68),(5,14),(5,15),(5,16),(5,17),(5,27),(5,33),(5,67),(6,10),(6,11),(6,12),(6,13),(6,26),(6,32),(6,67),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,66),(8,22),(8,23),(8,24),(8,25),(8,26),(8,27),(8,66),(9,66),(9,67),(9,68),(9,69),(10,34),(10,46),(10,58),(10,59),(10,87),(11,35),(11,47),(11,60),(11,61),(11,87),(12,36),(12,48),(12,58),(12,60),(12,88),(13,37),(13,49),(13,59),(13,61),(13,88),(14,38),(14,50),(14,62),(14,63),(14,87),(15,39),(15,51),(15,64),(15,65),(15,87),(16,40),(16,52),(16,62),(16,64),(16,88),(17,41),(17,53),(17,63),(17,65),(17,88),(18,42),(18,54),(18,58),(18,62),(18,86),(19,43),(19,55),(19,59),(19,63),(19,86),(20,44),(20,56),(20,60),(20,64),(20,86),(21,45),(21,57),(21,61),(21,65),(21,86),(22,34),(22,38),(22,42),(22,43),(22,89),(23,35),(23,39),(23,44),(23,45),(23,89),(24,36),(24,40),(24,42),(24,44),(24,90),(25,37),(25,41),(25,43),(25,45),(25,90),(26,34),(26,35),(26,36),(26,37),(26,91),(27,38),(27,39),(27,40),(27,41),(27,91),(28,46),(28,50),(28,54),(28,55),(28,89),(29,47),(29,51),(29,56),(29,57),(29,89),(30,48),(30,52),(30,54),(30,56),(30,90),(31,49),(31,53),(31,55),(31,57),(31,90),(32,46),(32,47),(32,48),(32,49),(32,91),(33,50),(33,51),(33,52),(33,53),(33,91),(34,70),(34,71),(34,93),(35,72),(35,73),(35,93),(36,70),(36,72),(36,94),(37,71),(37,73),(37,94),(38,74),(38,75),(38,93),(39,76),(39,77),(39,93),(40,74),(40,76),(40,94),(41,75),(41,77),(41,94),(42,70),(42,74),(42,95),(43,71),(43,75),(43,95),(44,72),(44,76),(44,95),(45,73),(45,77),(45,95),(46,78),(46,79),(46,93),(47,80),(47,81),(47,93),(48,78),(48,80),(48,94),(49,79),(49,81),(49,94),(50,82),(50,83),(50,93),(51,84),(51,85),(51,93),(52,82),(52,84),(52,94),(53,83),(53,85),(53,94),(54,78),(54,82),(54,95),(55,79),(55,83),(55,95),(56,80),(56,84),(56,95),(57,81),(57,85),(57,95),(58,70),(58,78),(58,92),(59,71),(59,79),(59,92),(60,72),(60,80),(60,92),(61,73),(61,81),(61,92),(62,74),(62,82),(62,92),(63,75),(63,83),(63,92),(64,76),(64,84),(64,92),(65,77),(65,85),(65,92),(66,89),(66,90),(66,91),(67,87),(67,88),(67,91),(68,86),(68,88),(68,90),(69,86),(69,87),(69,89),(70,96),(71,96),(72,96),(73,96),(74,96),(75,96),(76,96),(77,96),(78,96),(79,96),(80,96),(81,96),(82,96),(83,96),(84,96),(85,96),(86,92),(86,95),(87,92),(87,93),(88,92),(88,94),(89,93),(89,95),(90,94),(90,95),(91,93),(91,94),(92,96),(93,96),(94,96),(95,96)],97)
=> ? = 10
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,17),(1,21),(1,25),(1,29),(1,36),(1,37),(1,43),(2,12),(2,16),(2,20),(2,24),(2,28),(2,34),(2,35),(2,43),(3,15),(3,19),(3,23),(3,27),(3,31),(3,35),(3,37),(3,42),(4,14),(4,18),(4,22),(4,26),(4,30),(4,34),(4,36),(4,42),(5,11),(5,12),(5,13),(5,14),(5,15),(5,70),(5,71),(6,20),(6,21),(6,22),(6,23),(6,33),(6,69),(6,71),(7,16),(7,17),(7,18),(7,19),(7,33),(7,68),(7,70),(8,28),(8,29),(8,30),(8,31),(8,32),(8,68),(8,71),(9,24),(9,25),(9,26),(9,27),(9,32),(9,69),(9,70),(10,11),(10,42),(10,43),(10,68),(10,69),(11,80),(11,81),(11,103),(12,38),(12,39),(12,80),(12,82),(12,86),(13,40),(13,41),(13,80),(13,83),(13,87),(14,38),(14,40),(14,81),(14,84),(14,88),(15,39),(15,41),(15,81),(15,85),(15,89),(16,44),(16,45),(16,60),(16,82),(16,99),(17,46),(17,47),(17,61),(17,83),(17,99),(18,44),(18,46),(18,62),(18,84),(18,100),(19,45),(19,47),(19,63),(19,85),(19,100),(20,48),(20,49),(20,60),(20,86),(20,101),(21,50),(21,51),(21,61),(21,87),(21,101),(22,48),(22,50),(22,62),(22,88),(22,102),(23,49),(23,51),(23,63),(23,89),(23,102),(24,52),(24,53),(24,64),(24,82),(24,101),(25,54),(25,55),(25,65),(25,83),(25,101),(26,52),(26,54),(26,66),(26,84),(26,102),(27,53),(27,55),(27,67),(27,85),(27,102),(28,56),(28,57),(28,64),(28,86),(28,99),(29,58),(29,59),(29,65),(29,87),(29,99),(30,56),(30,58),(30,66),(30,88),(30,100),(31,57),(31,59),(31,67),(31,89),(31,100),(32,64),(32,65),(32,66),(32,67),(32,103),(33,60),(33,61),(33,62),(33,63),(33,103),(34,38),(34,44),(34,48),(34,52),(34,56),(34,98),(35,39),(35,45),(35,49),(35,53),(35,57),(35,98),(36,40),(36,46),(36,50),(36,54),(36,58),(36,98),(37,41),(37,47),(37,51),(37,55),(37,59),(37,98),(38,90),(38,94),(38,104),(39,91),(39,95),(39,104),(40,92),(40,96),(40,104),(41,93),(41,97),(41,104),(42,81),(42,98),(42,100),(42,102),(43,80),(43,98),(43,99),(43,101),(44,72),(44,90),(44,105),(45,73),(45,91),(45,105),(46,74),(46,92),(46,105),(47,75),(47,93),(47,105),(48,72),(48,94),(48,106),(49,73),(49,95),(49,106),(50,74),(50,96),(50,106),(51,75),(51,97),(51,106),(52,76),(52,90),(52,106),(53,77),(53,91),(53,106),(54,78),(54,92),(54,106),(55,79),(55,93),(55,106),(56,76),(56,94),(56,105),(57,77),(57,95),(57,105),(58,78),(58,96),(58,105),(59,79),(59,97),(59,105),(60,72),(60,73),(60,107),(61,74),(61,75),(61,107),(62,72),(62,74),(62,108),(63,73),(63,75),(63,108),(64,76),(64,77),(64,107),(65,78),(65,79),(65,107),(66,76),(66,78),(66,108),(67,77),(67,79),(67,108),(68,99),(68,100),(68,103),(69,101),(69,102),(69,103),(70,82),(70,83),(70,84),(70,85),(70,103),(71,86),(71,87),(71,88),(71,89),(71,103),(72,109),(73,109),(74,109),(75,109),(76,109),(77,109),(78,109),(79,109),(80,104),(80,107),(81,104),(81,108),(82,90),(82,91),(82,107),(83,92),(83,93),(83,107),(84,90),(84,92),(84,108),(85,91),(85,93),(85,108),(86,94),(86,95),(86,107),(87,96),(87,97),(87,107),(88,94),(88,96),(88,108),(89,95),(89,97),(89,108),(90,109),(91,109),(92,109),(93,109),(94,109),(95,109),(96,109),(97,109),(98,104),(98,105),(98,106),(99,105),(99,107),(100,105),(100,108),(101,106),(101,107),(102,106),(102,108),(103,107),(103,108),(104,109),(105,109),(106,109),(107,109),(108,109)],110)
=> ? = 10
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,17),(1,30),(1,31),(1,39),(1,40),(1,41),(1,42),(1,49),(2,14),(2,15),(2,28),(2,29),(2,35),(2,36),(2,37),(2,38),(2,49),(3,19),(3,23),(3,27),(3,34),(3,36),(3,40),(3,74),(3,79),(4,18),(4,22),(4,26),(4,33),(4,35),(4,39),(4,74),(4,78),(5,21),(5,22),(5,24),(5,34),(5,37),(5,41),(5,75),(5,76),(6,20),(6,23),(6,25),(6,33),(6,38),(6,42),(6,75),(6,77),(7,13),(7,20),(7,21),(7,29),(7,31),(7,32),(7,78),(7,79),(8,12),(8,18),(8,19),(8,28),(8,30),(8,32),(8,76),(8,77),(9,14),(9,16),(9,25),(9,26),(9,48),(9,76),(9,79),(10,15),(10,17),(10,24),(10,27),(10,48),(10,77),(10,78),(11,12),(11,13),(11,48),(11,49),(11,74),(11,75),(12,43),(12,84),(12,98),(12,121),(13,43),(13,85),(13,99),(13,122),(14,55),(14,56),(14,86),(14,89),(14,109),(15,54),(15,57),(15,87),(15,88),(15,109),(16,59),(16,60),(16,90),(16,93),(16,109),(17,58),(17,61),(17,91),(17,92),(17,109),(18,62),(18,66),(18,84),(18,94),(18,113),(19,63),(19,67),(19,84),(19,95),(19,114),(20,64),(20,68),(20,85),(20,97),(20,113),(21,65),(21,69),(21,85),(21,96),(21,114),(22,50),(22,52),(22,94),(22,96),(22,110),(23,51),(23,53),(23,95),(23,97),(23,110),(24,54),(24,58),(24,71),(24,96),(24,121),(25,55),(25,59),(25,70),(25,97),(25,121),(26,56),(26,60),(26,70),(26,94),(26,122),(27,57),(27,61),(27,71),(27,95),(27,122),(28,62),(28,63),(28,72),(28,86),(28,87),(28,98),(29,64),(29,65),(29,72),(29,88),(29,89),(29,99),(30,66),(30,67),(30,73),(30,90),(30,91),(30,98),(31,68),(31,69),(31,73),(31,92),(31,93),(31,99),(32,43),(32,72),(32,73),(32,113),(32,114),(33,44),(33,46),(33,70),(33,110),(33,113),(34,45),(34,47),(34,71),(34,110),(34,114),(35,44),(35,50),(35,56),(35,62),(35,88),(35,111),(36,45),(36,51),(36,57),(36,63),(36,89),(36,111),(37,45),(37,50),(37,54),(37,65),(37,86),(37,112),(38,44),(38,51),(38,55),(38,64),(38,87),(38,112),(39,46),(39,52),(39,60),(39,66),(39,92),(39,111),(40,47),(40,53),(40,61),(40,67),(40,93),(40,111),(41,47),(41,52),(41,58),(41,69),(41,90),(41,112),(42,46),(42,53),(42,59),(42,68),(42,91),(42,112),(43,100),(43,126),(44,80),(44,117),(44,123),(45,81),(45,118),(45,123),(46,82),(46,119),(46,123),(47,83),(47,120),(47,123),(48,109),(48,121),(48,122),(49,98),(49,99),(49,109),(49,111),(49,112),(50,101),(50,103),(50,123),(51,102),(51,104),(51,123),(52,105),(52,107),(52,123),(53,106),(53,108),(53,123),(54,81),(54,103),(54,124),(55,80),(55,104),(55,124),(56,80),(56,101),(56,125),(57,81),(57,102),(57,125),(58,83),(58,107),(58,124),(59,82),(59,108),(59,124),(60,82),(60,105),(60,125),(61,83),(61,106),(61,125),(62,101),(62,115),(62,117),(63,102),(63,115),(63,118),(64,104),(64,116),(64,117),(65,103),(65,116),(65,118),(66,105),(66,115),(66,119),(67,106),(67,115),(67,120),(68,108),(68,116),(68,119),(69,107),(69,116),(69,120),(70,80),(70,82),(70,126),(71,81),(71,83),(71,126),(72,100),(72,117),(72,118),(73,100),(73,119),(73,120),(74,84),(74,110),(74,111),(74,122),(75,85),(75,110),(75,112),(75,121),(76,86),(76,90),(76,94),(76,114),(76,121),(77,87),(77,91),(77,95),(77,113),(77,121),(78,88),(78,92),(78,96),(78,113),(78,122),(79,89),(79,93),(79,97),(79,114),(79,122),(80,127),(81,127),(82,127),(83,127),(84,115),(84,126),(85,116),(85,126),(86,101),(86,118),(86,124),(87,102),(87,117),(87,124),(88,103),(88,117),(88,125),(89,104),(89,118),(89,125),(90,105),(90,120),(90,124),(91,106),(91,119),(91,124),(92,107),(92,119),(92,125),(93,108),(93,120),(93,125),(94,101),(94,105),(94,126),(95,102),(95,106),(95,126),(96,103),(96,107),(96,126),(97,104),(97,108),(97,126),(98,100),(98,115),(98,124),(99,100),(99,116),(99,125),(100,127),(101,127),(102,127),(103,127),(104,127),(105,127),(106,127),(107,127),(108,127),(109,124),(109,125),(110,123),(110,126),(111,115),(111,123),(111,125),(112,116),(112,123),(112,124),(113,117),(113,119),(113,126),(114,118),(114,120),(114,126),(115,127),(116,127),(117,127),(118,127),(119,127),(120,127),(121,124),(121,126),(122,125),(122,126),(123,127),(124,127),(125,127),(126,127)],128)
=> ? = 20
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,19),(1,20),(1,21),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,46),(2,16),(2,17),(2,18),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(2,46),(3,15),(3,18),(3,21),(3,30),(3,33),(3,76),(3,79),(3,80),(4,14),(4,17),(4,20),(4,29),(4,32),(4,75),(4,78),(4,80),(5,13),(5,16),(5,19),(5,28),(5,31),(5,74),(5,77),(5,80),(6,23),(6,26),(6,28),(6,34),(6,40),(6,71),(6,75),(6,76),(7,22),(7,27),(7,29),(7,35),(7,41),(7,72),(7,74),(7,76),(8,24),(8,25),(8,30),(8,36),(8,42),(8,73),(8,74),(8,75),(9,25),(9,27),(9,31),(9,37),(9,43),(9,71),(9,78),(9,79),(10,24),(10,26),(10,32),(10,38),(10,44),(10,72),(10,77),(10,79),(11,22),(11,23),(11,33),(11,39),(11,45),(11,73),(11,77),(11,78),(12,13),(12,14),(12,15),(12,46),(12,71),(12,72),(12,73),(13,81),(13,84),(13,99),(13,127),(14,81),(14,85),(14,100),(14,128),(15,81),(15,86),(15,101),(15,129),(16,47),(16,50),(16,82),(16,84),(16,87),(16,90),(17,48),(17,51),(17,82),(17,85),(17,88),(17,91),(18,49),(18,52),(18,82),(18,86),(18,89),(18,92),(19,53),(19,56),(19,83),(19,84),(19,93),(19,96),(20,54),(20,57),(20,83),(20,85),(20,94),(20,97),(21,55),(21,58),(21,83),(21,86),(21,95),(21,98),(22,59),(22,65),(22,103),(22,107),(22,127),(23,60),(23,66),(23,102),(23,107),(23,128),(24,61),(24,67),(24,104),(24,106),(24,127),(25,62),(25,68),(25,104),(25,105),(25,128),(26,63),(26,69),(26,102),(26,106),(26,129),(27,64),(27,70),(27,103),(27,105),(27,129),(28,47),(28,53),(28,99),(28,102),(28,131),(29,48),(29,54),(29,100),(29,103),(29,131),(30,49),(30,55),(30,101),(30,104),(30,131),(31,50),(31,56),(31,99),(31,105),(31,130),(32,51),(32,57),(32,100),(32,106),(32,130),(33,52),(33,58),(33,101),(33,107),(33,130),(34,47),(34,60),(34,63),(34,88),(34,89),(34,120),(35,48),(35,59),(35,64),(35,87),(35,89),(35,121),(36,49),(36,61),(36,62),(36,87),(36,88),(36,122),(37,50),(37,62),(37,64),(37,91),(37,92),(37,120),(38,51),(38,61),(38,63),(38,90),(38,92),(38,121),(39,52),(39,59),(39,60),(39,90),(39,91),(39,122),(40,53),(40,66),(40,69),(40,94),(40,95),(40,120),(41,54),(41,65),(41,70),(41,93),(41,95),(41,121),(42,55),(42,67),(42,68),(42,93),(42,94),(42,122),(43,56),(43,68),(43,70),(43,97),(43,98),(43,120),(44,57),(44,67),(44,69),(44,96),(44,98),(44,121),(45,58),(45,65),(45,66),(45,96),(45,97),(45,122),(46,84),(46,85),(46,86),(46,120),(46,121),(46,122),(47,108),(47,124),(47,132),(48,109),(48,125),(48,132),(49,110),(49,126),(49,132),(50,111),(50,124),(50,133),(51,112),(51,125),(51,133),(52,113),(52,126),(52,133),(53,114),(53,124),(53,134),(54,115),(54,125),(54,134),(55,116),(55,126),(55,134),(56,117),(56,124),(56,135),(57,118),(57,125),(57,135),(58,119),(58,126),(58,135),(59,109),(59,113),(59,136),(60,108),(60,113),(60,137),(61,110),(61,112),(61,136),(62,110),(62,111),(62,137),(63,108),(63,112),(63,138),(64,109),(64,111),(64,138),(65,115),(65,119),(65,136),(66,114),(66,119),(66,137),(67,116),(67,118),(67,136),(68,116),(68,117),(68,137),(69,114),(69,118),(69,138),(70,115),(70,117),(70,138),(71,99),(71,120),(71,128),(71,129),(72,100),(72,121),(72,127),(72,129),(73,101),(73,122),(73,127),(73,128),(74,87),(74,93),(74,105),(74,127),(74,131),(75,88),(75,94),(75,106),(75,128),(75,131),(76,89),(76,95),(76,107),(76,129),(76,131),(77,90),(77,96),(77,102),(77,127),(77,130),(78,91),(78,97),(78,103),(78,128),(78,130),(79,92),(79,98),(79,104),(79,129),(79,130),(80,81),(80,82),(80,83),(80,130),(80,131),(81,123),(81,139),(82,123),(82,132),(82,133),(83,123),(83,134),(83,135),(84,123),(84,124),(84,136),(85,123),(85,125),(85,137),(86,123),(86,126),(86,138),(87,111),(87,132),(87,136),(88,112),(88,132),(88,137),(89,113),(89,132),(89,138),(90,108),(90,133),(90,136),(91,109),(91,133),(91,137),(92,110),(92,133),(92,138),(93,117),(93,134),(93,136),(94,118),(94,134),(94,137),(95,119),(95,134),(95,138),(96,114),(96,135),(96,136),(97,115),(97,135),(97,137),(98,116),(98,135),(98,138),(99,124),(99,139),(100,125),(100,139),(101,126),(101,139),(102,108),(102,114),(102,139),(103,109),(103,115),(103,139),(104,110),(104,116),(104,139),(105,111),(105,117),(105,139),(106,112),(106,118),(106,139),(107,113),(107,119),(107,139),(108,140),(109,140),(110,140),(111,140),(112,140),(113,140),(114,140),(115,140),(116,140),(117,140),(118,140),(119,140),(120,124),(120,137),(120,138),(121,125),(121,136),(121,138),(122,126),(122,136),(122,137),(123,140),(124,140),(125,140),(126,140),(127,136),(127,139),(128,137),(128,139),(129,138),(129,139),(130,133),(130,135),(130,139),(131,132),(131,134),(131,139),(132,140),(133,140),(134,140),(135,140),(136,140),(137,140),(138,140),(139,140)],141)
=> ? = 10
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,15),(1,26),(1,27),(1,32),(1,33),(1,41),(1,42),(1,91),(1,94),(2,14),(2,23),(2,25),(2,29),(2,31),(2,40),(2,42),(2,90),(2,93),(3,13),(3,22),(3,24),(3,28),(3,30),(3,40),(3,41),(3,89),(3,92),(4,18),(4,24),(4,25),(4,34),(4,35),(4,43),(4,44),(4,86),(4,94),(5,17),(5,22),(5,26),(5,36),(5,38),(5,43),(5,45),(5,87),(5,93),(6,16),(6,23),(6,27),(6,37),(6,39),(6,44),(6,45),(6,88),(6,92),(7,21),(7,30),(7,31),(7,36),(7,37),(7,46),(7,47),(7,86),(7,91),(8,20),(8,28),(8,32),(8,34),(8,39),(8,46),(8,48),(8,87),(8,90),(9,19),(9,29),(9,33),(9,35),(9,38),(9,47),(9,48),(9,88),(9,89),(10,19),(10,20),(10,21),(10,49),(10,92),(10,93),(10,94),(11,16),(11,17),(11,18),(11,49),(11,89),(11,90),(11,91),(12,13),(12,14),(12,15),(12,49),(12,86),(12,87),(12,88),(13,53),(13,54),(13,98),(13,100),(13,134),(14,53),(14,55),(14,99),(14,101),(14,135),(15,54),(15,55),(15,102),(15,103),(15,136),(16,56),(16,58),(16,107),(16,108),(16,134),(17,57),(17,58),(17,106),(17,109),(17,135),(18,56),(18,57),(18,104),(18,105),(18,136),(19,59),(19,61),(19,113),(19,114),(19,134),(20,60),(20,61),(20,112),(20,115),(20,135),(21,59),(21,60),(21,110),(21,111),(21,136),(22,63),(22,65),(22,80),(22,98),(22,106),(22,131),(23,64),(23,66),(23,81),(23,99),(23,107),(23,131),(24,62),(24,65),(24,82),(24,100),(24,104),(24,132),(25,62),(25,66),(25,83),(25,101),(25,105),(25,133),(26,63),(26,67),(26,84),(26,102),(26,109),(26,133),(27,64),(27,67),(27,85),(27,103),(27,108),(27,132),(28,69),(28,74),(28,82),(28,98),(28,112),(28,128),(29,70),(29,75),(29,83),(29,99),(29,113),(29,128),(30,68),(30,74),(30,80),(30,100),(30,110),(30,129),(31,68),(31,75),(31,81),(31,101),(31,111),(31,130),(32,69),(32,76),(32,85),(32,102),(32,115),(32,130),(33,70),(33,76),(33,84),(33,103),(33,114),(33,129),(34,72),(34,79),(34,82),(34,105),(34,115),(34,126),(35,71),(35,79),(35,83),(35,104),(35,114),(35,125),(36,73),(36,77),(36,80),(36,109),(36,111),(36,126),(37,73),(37,78),(37,81),(37,108),(37,110),(37,125),(38,71),(38,77),(38,84),(38,106),(38,113),(38,127),(39,72),(39,78),(39,85),(39,107),(39,112),(39,127),(40,50),(40,53),(40,62),(40,68),(40,128),(40,131),(41,50),(41,54),(41,63),(41,69),(41,129),(41,132),(42,50),(42,55),(42,64),(42,70),(42,130),(42,133),(43,51),(43,57),(43,65),(43,71),(43,126),(43,133),(44,51),(44,56),(44,66),(44,72),(44,125),(44,132),(45,51),(45,58),(45,67),(45,73),(45,127),(45,131),(46,52),(46,60),(46,74),(46,78),(46,126),(46,130),(47,52),(47,59),(47,75),(47,77),(47,125),(47,129),(48,52),(48,61),(48,76),(48,79),(48,127),(48,128),(49,134),(49,135),(49,136),(50,95),(50,147),(50,148),(51,96),(51,146),(51,148),(52,97),(52,146),(52,147),(53,95),(53,118),(53,151),(54,95),(54,116),(54,149),(55,95),(55,117),(55,150),(56,96),(56,120),(56,149),(57,96),(57,119),(57,150),(58,96),(58,121),(58,151),(59,97),(59,123),(59,149),(60,97),(60,122),(60,150),(61,97),(61,124),(61,151),(62,118),(62,142),(62,148),(63,116),(63,140),(63,148),(64,117),(64,141),(64,148),(65,119),(65,137),(65,148),(66,120),(66,138),(66,148),(67,121),(67,139),(67,148),(68,118),(68,143),(68,147),(69,116),(69,144),(69,147),(70,117),(70,145),(70,147),(71,119),(71,145),(71,146),(72,120),(72,144),(72,146),(73,121),(73,143),(73,146),(74,122),(74,137),(74,147),(75,123),(75,138),(75,147),(76,124),(76,139),(76,147),(77,123),(77,140),(77,146),(78,122),(78,141),(78,146),(79,124),(79,142),(79,146),(80,137),(80,140),(80,143),(81,138),(81,141),(81,143),(82,137),(82,142),(82,144),(83,138),(83,142),(83,145),(84,139),(84,140),(84,145),(85,139),(85,141),(85,144),(86,100),(86,101),(86,125),(86,126),(86,136),(87,98),(87,102),(87,126),(87,127),(87,135),(88,99),(88,103),(88,125),(88,127),(88,134),(89,104),(89,106),(89,128),(89,129),(89,134),(90,105),(90,107),(90,128),(90,130),(90,135),(91,108),(91,109),(91,129),(91,130),(91,136),(92,110),(92,112),(92,131),(92,132),(92,134),(93,111),(93,113),(93,131),(93,133),(93,135),(94,114),(94,115),(94,132),(94,133),(94,136),(95,152),(96,152),(97,152),(98,116),(98,137),(98,151),(99,117),(99,138),(99,151),(100,118),(100,137),(100,149),(101,118),(101,138),(101,150),(102,116),(102,139),(102,150),(103,117),(103,139),(103,149),(104,119),(104,142),(104,149),(105,120),(105,142),(105,150),(106,119),(106,140),(106,151),(107,120),(107,141),(107,151),(108,121),(108,141),(108,149),(109,121),(109,140),(109,150),(110,122),(110,143),(110,149),(111,123),(111,143),(111,150),(112,122),(112,144),(112,151),(113,123),(113,145),(113,151),(114,124),(114,145),(114,149),(115,124),(115,144),(115,150),(116,152),(117,152),(118,152),(119,152),(120,152),(121,152),(122,152),(123,152),(124,152),(125,138),(125,146),(125,149),(126,137),(126,146),(126,150),(127,139),(127,146),(127,151),(128,142),(128,147),(128,151),(129,140),(129,147),(129,149),(130,141),(130,147),(130,150),(131,143),(131,148),(131,151),(132,144),(132,148),(132,149),(133,145),(133,148),(133,150),(134,149),(134,151),(135,150),(135,151),(136,149),(136,150),(137,152),(138,152),(139,152),(140,152),(141,152),(142,152),(143,152),(144,152),(145,152),(146,152),(147,152),(148,152),(149,152),(150,152),(151,152)],153)
=> ? = 10
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,25),(1,30),(1,31),(1,36),(1,37),(1,40),(1,43),(1,46),(1,64),(1,65),(2,24),(2,27),(2,29),(2,33),(2,35),(2,39),(2,42),(2,45),(2,63),(2,65),(3,23),(3,26),(3,28),(3,32),(3,34),(3,38),(3,41),(3,44),(3,63),(3,64),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,90),(4,91),(4,92),(5,16),(5,28),(5,29),(5,43),(5,47),(5,48),(5,84),(5,85),(5,90),(6,14),(6,26),(6,30),(6,42),(6,49),(6,51),(6,84),(6,86),(6,91),(7,15),(7,27),(7,31),(7,41),(7,50),(7,52),(7,85),(7,86),(7,92),(8,19),(8,34),(8,35),(8,46),(8,49),(8,50),(8,87),(8,88),(8,90),(9,17),(9,32),(9,36),(9,45),(9,47),(9,52),(9,87),(9,89),(9,91),(10,18),(10,33),(10,37),(10,44),(10,48),(10,51),(10,88),(10,89),(10,92),(11,15),(11,18),(11,20),(11,38),(11,62),(11,65),(11,84),(11,87),(12,14),(12,17),(12,21),(12,39),(12,62),(12,64),(12,85),(12,88),(13,16),(13,19),(13,22),(13,40),(13,62),(13,63),(13,86),(13,89),(14,67),(14,94),(14,97),(14,112),(14,146),(15,66),(15,93),(15,98),(15,113),(15,146),(16,68),(16,95),(16,96),(16,114),(16,146),(17,70),(17,94),(17,100),(17,116),(17,145),(18,69),(18,93),(18,101),(18,115),(18,145),(19,71),(19,95),(19,99),(19,117),(19,145),(20,59),(20,93),(20,102),(20,111),(20,148),(21,60),(21,94),(21,102),(21,110),(21,149),(22,61),(22,95),(22,102),(22,109),(22,150),(23,59),(23,103),(23,105),(23,109),(23,110),(23,124),(24,60),(24,104),(24,106),(24,109),(24,111),(24,125),(25,61),(25,107),(25,108),(25,110),(25,111),(25,126),(26,53),(26,80),(26,97),(26,103),(26,118),(26,139),(27,54),(27,81),(27,98),(27,104),(27,119),(27,139),(28,55),(28,78),(28,96),(28,105),(28,118),(28,140),(29,56),(29,79),(29,96),(29,106),(29,119),(29,141),(30,57),(30,83),(30,97),(30,107),(30,120),(30,141),(31,58),(31,82),(31,98),(31,108),(31,120),(31,140),(32,55),(32,74),(32,100),(32,103),(32,121),(32,142),(33,56),(33,75),(33,101),(33,104),(33,122),(33,142),(34,53),(34,72),(34,99),(34,105),(34,121),(34,143),(35,54),(35,73),(35,99),(35,106),(35,122),(35,144),(36,58),(36,77),(36,100),(36,107),(36,123),(36,144),(37,57),(37,76),(37,101),(37,108),(37,123),(37,143),(38,59),(38,66),(38,69),(38,118),(38,121),(38,147),(39,60),(39,67),(39,70),(39,119),(39,122),(39,147),(40,61),(40,68),(40,71),(40,120),(40,123),(40,147),(41,66),(41,72),(41,74),(41,124),(41,139),(41,140),(42,67),(42,73),(42,75),(42,125),(42,139),(42,141),(43,68),(43,76),(43,77),(43,126),(43,140),(43,141),(44,69),(44,78),(44,80),(44,124),(44,142),(44,143),(45,70),(45,79),(45,81),(45,125),(45,142),(45,144),(46,71),(46,82),(46,83),(46,126),(46,143),(46,144),(47,55),(47,77),(47,79),(47,114),(47,116),(47,148),(48,56),(48,76),(48,78),(48,114),(48,115),(48,149),(49,53),(49,73),(49,83),(49,112),(49,117),(49,148),(50,54),(50,72),(50,82),(50,113),(50,117),(50,149),(51,57),(51,75),(51,80),(51,112),(51,115),(51,150),(52,58),(52,74),(52,81),(52,113),(52,116),(52,150),(53,152),(53,154),(53,158),(54,153),(54,154),(54,159),(55,151),(55,155),(55,158),(56,151),(56,156),(56,159),(57,152),(57,156),(57,160),(58,153),(58,155),(58,160),(59,127),(59,157),(59,158),(60,128),(60,157),(60,159),(61,129),(61,157),(61,160),(62,102),(62,145),(62,146),(62,147),(63,96),(63,99),(63,109),(63,139),(63,142),(63,147),(64,97),(64,100),(64,110),(64,140),(64,143),(64,147),(65,98),(65,101),(65,111),(65,141),(65,144),(65,147),(66,127),(66,130),(66,164),(67,128),(67,131),(67,164),(68,129),(68,132),(68,164),(69,127),(69,133),(69,165),(70,128),(70,134),(70,165),(71,129),(71,135),(71,165),(72,130),(72,154),(72,162),(73,131),(73,154),(73,163),(74,130),(74,155),(74,161),(75,131),(75,156),(75,161),(76,132),(76,156),(76,162),(77,132),(77,155),(77,163),(78,133),(78,151),(78,162),(79,134),(79,151),(79,163),(80,133),(80,152),(80,161),(81,134),(81,153),(81,161),(82,135),(82,153),(82,162),(83,135),(83,152),(83,163),(84,115),(84,118),(84,141),(84,146),(84,148),(85,116),(85,119),(85,140),(85,146),(85,149),(86,117),(86,120),(86,139),(86,146),(86,150),(87,113),(87,121),(87,144),(87,145),(87,148),(88,112),(88,122),(88,143),(88,145),(88,149),(89,114),(89,123),(89,142),(89,145),(89,150),(90,95),(90,105),(90,106),(90,126),(90,148),(90,149),(91,94),(91,103),(91,107),(91,125),(91,148),(91,150),(92,93),(92,104),(92,108),(92,124),(92,149),(92,150),(93,127),(93,138),(93,166),(94,128),(94,137),(94,166),(95,129),(95,136),(95,166),(96,136),(96,151),(96,164),(97,137),(97,152),(97,164),(98,138),(98,153),(98,164),(99,136),(99,154),(99,165),(100,137),(100,155),(100,165),(101,138),(101,156),(101,165),(102,157),(102,166),(103,137),(103,158),(103,161),(104,138),(104,159),(104,161),(105,136),(105,158),(105,162),(106,136),(106,159),(106,163),(107,137),(107,160),(107,163),(108,138),(108,160),(108,162),(109,136),(109,157),(109,161),(110,137),(110,157),(110,162),(111,138),(111,157),(111,163),(112,131),(112,152),(112,166),(113,130),(113,153),(113,166),(114,132),(114,151),(114,166),(115,133),(115,156),(115,166),(116,134),(116,155),(116,166),(117,135),(117,154),(117,166),(118,133),(118,158),(118,164),(119,134),(119,159),(119,164),(120,135),(120,160),(120,164),(121,130),(121,158),(121,165),(122,131),(122,159),(122,165),(123,132),(123,160),(123,165),(124,127),(124,161),(124,162),(125,128),(125,161),(125,163),(126,129),(126,162),(126,163),(127,167),(128,167),(129,167),(130,167),(131,167),(132,167),(133,167),(134,167),(135,167),(136,167),(137,167),(138,167),(139,154),(139,161),(139,164),(140,155),(140,162),(140,164),(141,156),(141,163),(141,164),(142,151),(142,161),(142,165),(143,152),(143,162),(143,165),(144,153),(144,163),(144,165),(145,165),(145,166),(146,164),(146,166),(147,157),(147,164),(147,165),(148,158),(148,163),(148,166),(149,159),(149,162),(149,166),(150,160),(150,161),(150,166),(151,167),(152,167),(153,167),(154,167),(155,167),(156,167),(157,167),(158,167),(159,167),(160,167),(161,167),(162,167),(163,167),(164,167),(165,167),(166,167)],168)
=> ? = 10
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,16),(1,21),(1,22),(1,28),(1,47),(1,52),(1,53),(1,88),(1,89),(1,91),(2,15),(2,19),(2,20),(2,27),(2,46),(2,50),(2,51),(2,86),(2,87),(2,91),(3,18),(3,24),(3,26),(3,30),(3,49),(3,55),(3,57),(3,87),(3,89),(3,90),(4,17),(4,23),(4,25),(4,29),(4,48),(4,54),(4,56),(4,86),(4,88),(4,90),(5,15),(5,31),(5,32),(5,39),(5,47),(5,54),(5,55),(5,92),(5,93),(5,97),(6,16),(6,33),(6,34),(6,40),(6,46),(6,56),(6,57),(6,94),(6,95),(6,97),(7,17),(7,35),(7,37),(7,41),(7,49),(7,50),(7,52),(7,92),(7,94),(7,96),(8,18),(8,36),(8,38),(8,42),(8,48),(8,51),(8,53),(8,93),(8,95),(8,96),(9,19),(9,23),(9,31),(9,35),(9,44),(9,82),(9,84),(9,89),(9,95),(10,20),(10,24),(10,32),(10,36),(10,45),(10,82),(10,85),(10,88),(10,94),(11,21),(11,25),(11,33),(11,37),(11,45),(11,83),(11,84),(11,87),(11,93),(12,22),(12,26),(12,34),(12,38),(12,44),(12,83),(12,85),(12,86),(12,92),(13,29),(13,30),(13,41),(13,42),(13,43),(13,84),(13,85),(13,91),(13,97),(14,27),(14,28),(14,39),(14,40),(14,43),(14,82),(14,83),(14,90),(14,96),(15,70),(15,71),(15,78),(15,152),(15,153),(15,157),(16,72),(16,73),(16,79),(16,154),(16,155),(16,157),(17,74),(17,76),(17,80),(17,152),(17,154),(17,156),(18,75),(18,77),(18,81),(18,153),(18,155),(18,156),(19,58),(19,70),(19,99),(19,122),(19,134),(19,168),(20,59),(20,71),(20,98),(20,123),(20,134),(20,167),(21,60),(21,72),(21,101),(21,124),(21,135),(21,168),(22,61),(22,73),(22,100),(22,125),(22,135),(22,167),(23,62),(23,74),(23,104),(23,122),(23,136),(23,166),(24,63),(24,75),(24,105),(24,123),(24,137),(24,166),(25,64),(25,76),(25,102),(25,124),(25,136),(25,165),(26,65),(26,77),(26,103),(26,125),(26,137),(26,165),(27,66),(27,78),(27,106),(27,126),(27,134),(27,165),(28,67),(28,79),(28,107),(28,126),(28,135),(28,166),(29,68),(29,80),(29,108),(29,127),(29,136),(29,167),(30,69),(30,81),(30,109),(30,127),(30,137),(30,168),(31,62),(31,70),(31,113),(31,128),(31,138),(31,172),(32,63),(32,71),(32,112),(32,129),(32,138),(32,171),(33,64),(33,72),(33,111),(33,130),(33,139),(33,172),(34,65),(34,73),(34,110),(34,131),(34,139),(34,171),(35,58),(35,74),(35,117),(35,128),(35,140),(35,170),(36,59),(36,75),(36,116),(36,129),(36,141),(36,170),(37,60),(37,76),(37,115),(37,130),(37,140),(37,169),(38,61),(38,77),(38,114),(38,131),(38,141),(38,169),(39,67),(39,78),(39,118),(39,132),(39,138),(39,169),(40,66),(40,79),(40,119),(40,132),(40,139),(40,170),(41,69),(41,80),(41,120),(41,133),(41,140),(41,171),(42,68),(42,81),(42,121),(42,133),(42,141),(42,172),(43,126),(43,127),(43,132),(43,133),(43,158),(44,122),(44,125),(44,128),(44,131),(44,158),(45,123),(45,124),(45,129),(45,130),(45,158),(46,66),(46,98),(46,99),(46,110),(46,111),(46,157),(47,67),(47,100),(47,101),(47,112),(47,113),(47,157),(48,68),(48,102),(48,104),(48,114),(48,116),(48,156),(49,69),(49,103),(49,105),(49,115),(49,117),(49,156),(50,58),(50,98),(50,106),(50,115),(50,120),(50,152),(51,59),(51,99),(51,106),(51,114),(51,121),(51,153),(52,60),(52,100),(52,107),(52,117),(52,120),(52,154),(53,61),(53,101),(53,107),(53,116),(53,121),(53,155),(54,62),(54,102),(54,108),(54,112),(54,118),(54,152),(55,63),(55,103),(55,109),(55,113),(55,118),(55,153),(56,64),(56,104),(56,108),(56,110),(56,119),(56,154),(57,65),(57,105),(57,109),(57,111),(57,119),(57,155),(58,159),(58,173),(58,180),(59,160),(59,173),(59,179),(60,161),(60,174),(60,180),(61,162),(61,174),(61,179),(62,159),(62,175),(62,178),(63,160),(63,176),(63,178),(64,161),(64,175),(64,177),(65,162),(65,176),(65,177),(66,163),(66,173),(66,177),(67,163),(67,174),(67,178),(68,164),(68,175),(68,179),(69,164),(69,176),(69,180),(70,142),(70,159),(70,184),(71,142),(71,160),(71,183),(72,143),(72,161),(72,184),(73,143),(73,162),(73,183),(74,144),(74,159),(74,182),(75,145),(75,160),(75,182),(76,144),(76,161),(76,181),(77,145),(77,162),(77,181),(78,142),(78,163),(78,181),(79,143),(79,163),(79,182),(80,144),(80,164),(80,183),(81,145),(81,164),(81,184),(82,134),(82,138),(82,158),(82,166),(82,170),(83,135),(83,139),(83,158),(83,165),(83,169),(84,136),(84,140),(84,158),(84,168),(84,172),(85,137),(85,141),(85,158),(85,167),(85,171),(86,110),(86,114),(86,122),(86,152),(86,165),(86,167),(87,111),(87,115),(87,123),(87,153),(87,165),(87,168),(88,112),(88,116),(88,124),(88,154),(88,166),(88,167),(89,113),(89,117),(89,125),(89,155),(89,166),(89,168),(90,118),(90,119),(90,127),(90,156),(90,165),(90,166),(91,120),(91,121),(91,126),(91,157),(91,167),(91,168),(92,100),(92,103),(92,128),(92,152),(92,169),(92,171),(93,101),(93,102),(93,129),(93,153),(93,169),(93,172),(94,98),(94,105),(94,130),(94,154),(94,170),(94,171),(95,99),(95,104),(95,131),(95,155),(95,170),(95,172),(96,106),(96,107),(96,133),(96,156),(96,169),(96,170),(97,108),(97,109),(97,132),(97,157),(97,171),(97,172),(98,147),(98,173),(98,183),(99,146),(99,173),(99,184),(100,149),(100,174),(100,183),(101,148),(101,174),(101,184),(102,148),(102,175),(102,181),(103,149),(103,176),(103,181),(104,146),(104,175),(104,182),(105,147),(105,176),(105,182),(106,150),(106,173),(106,181),(107,150),(107,174),(107,182),(108,151),(108,175),(108,183),(109,151),(109,176),(109,184),(110,146),(110,177),(110,183),(111,147),(111,177),(111,184),(112,148),(112,178),(112,183),(113,149),(113,178),(113,184),(114,146),(114,179),(114,181),(115,147),(115,180),(115,181),(116,148),(116,179),(116,182),(117,149),(117,180),(117,182),(118,151),(118,178),(118,181),(119,151),(119,177),(119,182),(120,150),(120,180),(120,183),(121,150),(121,179),(121,184),(122,146),(122,159),(122,186),(123,147),(123,160),(123,186),(124,148),(124,161),(124,186),(125,149),(125,162),(125,186),(126,150),(126,163),(126,186),(127,151),(127,164),(127,186),(128,149),(128,159),(128,185),(129,148),(129,160),(129,185),(130,147),(130,161),(130,185),(131,146),(131,162),(131,185),(132,151),(132,163),(132,185),(133,150),(133,164),(133,185),(134,142),(134,173),(134,186),(135,143),(135,174),(135,186),(136,144),(136,175),(136,186),(137,145),(137,176),(137,186),(138,142),(138,178),(138,185),(139,143),(139,177),(139,185),(140,144),(140,180),(140,185),(141,145),(141,179),(141,185),(142,187),(143,187),(144,187),(145,187),(146,187),(147,187),(148,187),(149,187),(150,187),(151,187),(152,159),(152,181),(152,183),(153,160),(153,181),(153,184),(154,161),(154,182),(154,183),(155,162),(155,182),(155,184),(156,164),(156,181),(156,182),(157,163),(157,183),(157,184),(158,185),(158,186),(159,187),(160,187),(161,187),(162,187),(163,187),(164,187),(165,177),(165,181),(165,186),(166,178),(166,182),(166,186),(167,179),(167,183),(167,186),(168,180),(168,184),(168,186),(169,174),(169,181),(169,185),(170,173),(170,182),(170,185),(171,176),(171,183),(171,185),(172,175),(172,184),(172,185),(173,187),(174,187),(175,187),(176,187),(177,187),(178,187),(179,187),(180,187),(181,187),(182,187),(183,187),(184,187),(185,187),(186,187)],188)
=> ? = 5
[6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,82),(1,83),(1,84),(1,85),(2,18),(2,19),(2,25),(2,30),(2,31),(2,37),(2,78),(2,79),(2,81),(2,83),(3,16),(3,17),(3,24),(3,28),(3,29),(3,36),(3,76),(3,77),(3,81),(3,82),(4,21),(4,23),(4,27),(4,33),(4,35),(4,39),(4,77),(4,79),(4,80),(4,85),(5,20),(5,22),(5,26),(5,32),(5,34),(5,38),(5,76),(5,78),(5,80),(5,84),(6,22),(6,23),(6,24),(6,46),(6,47),(6,54),(6,83),(6,86),(6,87),(6,91),(7,20),(7,21),(7,25),(7,48),(7,49),(7,55),(7,82),(7,88),(7,89),(7,91),(8,17),(8,19),(8,26),(8,50),(8,52),(8,56),(8,85),(8,86),(8,88),(8,90),(9,16),(9,18),(9,27),(9,51),(9,53),(9,57),(9,84),(9,87),(9,89),(9,90),(10,28),(10,32),(10,43),(10,48),(10,51),(10,59),(10,79),(10,86),(10,92),(10,94),(11,29),(11,33),(11,42),(11,49),(11,50),(11,60),(11,78),(11,87),(11,92),(11,95),(12,30),(12,34),(12,41),(12,46),(12,53),(12,60),(12,77),(12,88),(12,93),(12,94),(13,31),(13,35),(13,40),(13,47),(13,52),(13,59),(13,76),(13,89),(13,93),(13,95),(14,38),(14,39),(14,45),(14,56),(14,57),(14,58),(14,81),(14,91),(14,94),(14,95),(15,36),(15,37),(15,44),(15,54),(15,55),(15,58),(15,80),(15,90),(15,92),(15,93),(16,61),(16,107),(16,112),(16,126),(16,134),(16,170),(17,62),(17,106),(17,113),(17,127),(17,134),(17,171),(18,63),(18,109),(18,112),(18,129),(18,135),(18,172),(19,64),(19,108),(19,113),(19,128),(19,135),(19,173),(20,65),(20,104),(20,110),(20,132),(20,136),(20,170),(21,66),(21,105),(21,110),(21,133),(21,137),(21,171),(22,67),(22,102),(22,111),(22,130),(22,136),(22,172),(23,68),(23,103),(23,111),(23,131),(23,137),(23,173),(24,69),(24,102),(24,103),(24,126),(24,127),(24,175),(25,70),(25,104),(25,105),(25,128),(25,129),(25,175),(26,71),(26,106),(26,108),(26,130),(26,132),(26,174),(27,72),(27,107),(27,109),(27,131),(27,133),(27,174),(28,61),(28,96),(28,114),(28,127),(28,138),(28,169),(29,62),(29,97),(29,115),(29,126),(29,138),(29,168),(30,63),(30,98),(30,117),(30,128),(30,139),(30,169),(31,64),(31,99),(31,116),(31,129),(31,139),(31,168),(32,65),(32,96),(32,118),(32,130),(32,140),(32,167),(33,66),(33,97),(33,119),(33,131),(33,141),(33,167),(34,67),(34,98),(34,120),(34,132),(34,140),(34,166),(35,68),(35,99),(35,121),(35,133),(35,141),(35,166),(36,69),(36,100),(36,122),(36,134),(36,138),(36,166),(37,70),(37,100),(37,123),(37,135),(37,139),(37,167),(38,71),(38,101),(38,124),(38,136),(38,140),(38,168),(39,72),(39,101),(39,125),(39,137),(39,141),(39,169),(40,73),(40,116),(40,121),(40,142),(40,144),(40,170),(41,74),(41,117),(41,120),(41,142),(41,145),(41,171),(42,74),(42,115),(42,119),(42,143),(42,144),(42,172),(43,73),(43,114),(43,118),(43,143),(43,145),(43,173),(44,75),(44,122),(44,123),(44,142),(44,143),(44,174),(45,75),(45,124),(45,125),(45,144),(45,145),(45,175),(46,67),(46,103),(46,117),(46,148),(46,152),(46,178),(47,68),(47,102),(47,116),(47,149),(47,152),(47,179),(48,65),(48,105),(48,114),(48,146),(48,153),(48,178),(49,66),(49,104),(49,115),(49,147),(49,153),(49,179),(50,62),(50,108),(50,119),(50,147),(50,154),(50,176),(51,61),(51,109),(51,118),(51,146),(51,155),(51,176),(52,64),(52,106),(52,121),(52,149),(52,154),(52,177),(53,63),(53,107),(53,120),(53,148),(53,155),(53,177),(54,69),(54,111),(54,123),(54,150),(54,152),(54,176),(55,70),(55,110),(55,122),(55,150),(55,153),(55,177),(56,71),(56,113),(56,125),(56,151),(56,154),(56,178),(57,72),(57,112),(57,124),(57,151),(57,155),(57,179),(58,75),(58,100),(58,101),(58,150),(58,151),(58,180),(59,73),(59,96),(59,99),(59,146),(59,149),(59,180),(60,74),(60,97),(60,98),(60,147),(60,148),(60,180),(61,181),(61,189),(61,190),(62,181),(62,188),(62,191),(63,182),(63,189),(63,192),(64,182),(64,188),(64,193),(65,183),(65,187),(65,190),(66,184),(66,187),(66,191),(67,183),(67,186),(67,192),(68,184),(68,186),(68,193),(69,181),(69,186),(69,194),(70,182),(70,187),(70,194),(71,183),(71,188),(71,195),(72,184),(72,189),(72,195),(73,185),(73,190),(73,193),(74,185),(74,191),(74,192),(75,185),(75,194),(75,195),(76,96),(76,102),(76,106),(76,166),(76,168),(76,170),(77,97),(77,103),(77,107),(77,166),(77,169),(77,171),(78,98),(78,104),(78,108),(78,167),(78,168),(78,172),(79,99),(79,105),(79,109),(79,167),(79,169),(79,173),(80,101),(80,110),(80,111),(80,166),(80,167),(80,174),(81,100),(81,112),(81,113),(81,168),(81,169),(81,175),(82,114),(82,115),(82,122),(82,170),(82,171),(82,175),(83,116),(83,117),(83,123),(83,172),(83,173),(83,175),(84,118),(84,120),(84,124),(84,170),(84,172),(84,174),(85,119),(85,121),(85,125),(85,171),(85,173),(85,174),(86,127),(86,130),(86,149),(86,173),(86,176),(86,178),(87,126),(87,131),(87,148),(87,172),(87,176),(87,179),(88,128),(88,132),(88,147),(88,171),(88,177),(88,178),(89,129),(89,133),(89,146),(89,170),(89,177),(89,179),(90,134),(90,135),(90,151),(90,174),(90,176),(90,177),(91,136),(91,137),(91,150),(91,175),(91,178),(91,179),(92,138),(92,143),(92,153),(92,167),(92,176),(92,180),(93,139),(93,142),(93,152),(93,166),(93,177),(93,180),(94,140),(94,145),(94,155),(94,169),(94,178),(94,180),(95,141),(95,144),(95,154),(95,168),(95,179),(95,180),(96,156),(96,190),(96,197),(97,157),(97,191),(97,197),(98,158),(98,192),(98,197),(99,159),(99,193),(99,197),(100,160),(100,194),(100,197),(101,161),(101,195),(101,197),(102,156),(102,186),(102,200),(103,157),(103,186),(103,201),(104,158),(104,187),(104,200),(105,159),(105,187),(105,201),(106,156),(106,188),(106,198),(107,157),(107,189),(107,198),(108,158),(108,188),(108,199),(109,159),(109,189),(109,199),(110,161),(110,187),(110,198),(111,161),(111,186),(111,199),(112,160),(112,189),(112,200),(113,160),(113,188),(113,201),(114,162),(114,190),(114,201),(115,162),(115,191),(115,200),(116,163),(116,193),(116,200),(117,163),(117,192),(117,201),(118,164),(118,190),(118,199),(119,165),(119,191),(119,199),(120,164),(120,192),(120,198),(121,165),(121,193),(121,198),(122,162),(122,194),(122,198),(123,163),(123,194),(123,199),(124,164),(124,195),(124,200),(125,165),(125,195),(125,201),(126,157),(126,181),(126,200),(127,156),(127,181),(127,201),(128,158),(128,182),(128,201),(129,159),(129,182),(129,200),(130,156),(130,183),(130,199),(131,157),(131,184),(131,199),(132,158),(132,183),(132,198),(133,159),(133,184),(133,198),(134,160),(134,181),(134,198),(135,160),(135,182),(135,199),(136,161),(136,183),(136,200),(137,161),(137,184),(137,201),(138,162),(138,181),(138,197),(139,163),(139,182),(139,197),(140,164),(140,183),(140,197),(141,165),(141,184),(141,197),(142,163),(142,185),(142,198),(143,162),(143,185),(143,199),(144,165),(144,185),(144,200),(145,164),(145,185),(145,201),(146,159),(146,190),(146,196),(147,158),(147,191),(147,196),(148,157),(148,192),(148,196),(149,156),(149,193),(149,196),(150,161),(150,194),(150,196),(151,160),(151,195),(151,196),(152,163),(152,186),(152,196),(153,162),(153,187),(153,196),(154,165),(154,188),(154,196),(155,164),(155,189),(155,196),(156,202),(157,202),(158,202),(159,202),(160,202),(161,202),(162,202),(163,202),(164,202),(165,202),(166,186),(166,197),(166,198),(167,187),(167,197),(167,199),(168,188),(168,197),(168,200),(169,189),(169,197),(169,201),(170,190),(170,198),(170,200),(171,191),(171,198),(171,201),(172,192),(172,199),(172,200),(173,193),(173,199),(173,201),(174,195),(174,198),(174,199),(175,194),(175,200),(175,201),(176,181),(176,196),(176,199),(177,182),(177,196),(177,198),(178,183),(178,196),(178,201),(179,184),(179,196),(179,200),(180,185),(180,196),(180,197),(181,202),(182,202),(183,202),(184,202),(185,202),(186,202),(187,202),(188,202),(189,202),(190,202),(191,202),(192,202),(193,202),(194,202),(195,202),(196,202),(197,202),(198,202),(199,202),(200,202),(201,202)],203)
=> ? = 1
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([],1)
=> 1
[1,1,1,1,1,2] => [1,6] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3
[1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 4
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 4
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 8
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 6
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 6
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
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