Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001074
(load all 3 compositions to match this statistic)
Mapping
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 5
[2,1,3] => 3
[2,3,1] => 5
[3,1,2] => 3
[3,2,1] => 3
[1,2,3,4] => 4
[1,2,4,3] => 6
[1,3,2,4] => 6
[1,3,4,2] => 8
[1,4,2,3] => 8
[1,4,3,2] => 8
[2,1,3,4] => 4
[2,1,4,3] => 6
[2,3,1,4] => 6
[2,3,4,1] => 8
[2,4,1,3] => 6
[2,4,3,1] => 8
[3,1,2,4] => 4
[3,1,4,2] => 6
[3,2,1,4] => 4
[3,2,4,1] => 8
[3,4,1,2] => 6
[3,4,2,1] => 6
[4,1,2,3] => 4
[4,1,3,2] => 6
[4,2,1,3] => 4
[4,2,3,1] => 6
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 5
[1,2,3,5,4] => 7
[1,2,4,3,5] => 7
[1,2,4,5,3] => 9
[1,2,5,3,4] => 9
[1,2,5,4,3] => 9
[1,3,2,4,5] => 7
[1,3,2,5,4] => 9
[1,3,4,2,5] => 9
[1,3,4,5,2] => 11
[1,3,5,2,4] => 9
[1,3,5,4,2] => 11
[1,4,2,3,5] => 9
[1,4,2,5,3] => 11
[1,4,3,2,5] => 9
[1,4,3,5,2] => 13
[1,4,5,2,3] => 11
[1,4,5,3,2] => 11
Description
The number of inversions of the cyclic embedding of a permutation. The cyclic embedding of a permutation $\pi$ of length $n$ is given by the permutation of length $n+1$ represented in cycle notation by $(\pi_1,\ldots,\pi_n,n+1)$. This reflects in particular the fact that the number of long cycles of length $n+1$ equals $n!$. This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length $n$ equals $n!\cdot(3n-1)/12$.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2
[2,1] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2
[1,2,3] => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 5
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 5
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 7
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 7
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 9
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 7
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 9
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 11
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11
[1,4,5,3,2] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11
[1,5,2,3,4] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11
[1,5,2,4,3] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13
[1,5,3,2,4] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11
[1,5,3,4,2] => [5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13
[1,5,4,2,3] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11
[1,5,4,3,2] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11
[2,1,3,4,5] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,1,3,5,4] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 7
[2,1,4,3,5] => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 7
[2,1,4,5,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 9
[2,1,5,3,4] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9
[2,1,5,4,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9
[2,3,1,4,5] => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 7
[2,3,1,5,4] => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 9
[3,1,2,4,5] => [3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,2,1,4,5] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1,2,3,5] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,2,1,3,5] => [2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,3,1,2,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,3,2,1,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,1,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,2,1,3,4] => [1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,1,2,4] => [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,2,1,4] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,1,2,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,2,1,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,1,3,4,5,6] => [5,6,4,3,2,1] => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[3,1,2,4,5,6] => [4,6,5,3,2,1] => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[3,2,1,4,5,6] => [4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,1,2,3,5,6] => [3,6,5,4,2,1] => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,2,1,3,5,6] => [3,5,6,4,2,1] => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,3,1,2,5,6] => [3,4,6,5,2,1] => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,3,2,1,5,6] => [3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,1,2,3,4,6] => [2,6,5,4,3,1] => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,2,1,3,4,6] => [2,5,6,4,3,1] => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,3,1,2,4,6] => [2,4,6,5,3,1] => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,3,2,1,4,6] => [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,4,1,2,3,6] => [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,4,2,1,3,6] => [2,3,5,6,4,1] => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,4,3,1,2,6] => [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,4,3,2,1,6] => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,2,1,3,4,5] => [1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,3,1,2,4,5] => [1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,3,2,1,4,5] => [1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,4,1,2,3,5] => [1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,4,2,1,3,5] => [1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 5 - 1
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 5 - 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6 - 1
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6 - 1
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 6 - 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6 - 1
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 6 - 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 7 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 7 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 9 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 7 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 9 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 11 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 9 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11 - 1
[1,4,5,3,2] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 11 - 1
[1,5,2,3,4] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13 - 1
[1,5,3,2,4] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11 - 1
[1,5,3,4,2] => [5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 13 - 1
[1,5,4,2,3] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11 - 1
[1,5,4,3,2] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 11 - 1
[2,1,3,4,5] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,1,3,5,4] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 7 - 1
[2,1,4,3,5] => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 7 - 1
[2,1,4,5,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 9 - 1
[2,1,5,3,4] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9 - 1
[2,1,5,4,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 9 - 1
[2,3,1,4,5] => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 7 - 1
[2,3,1,5,4] => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 9 - 1
[3,1,2,4,5] => [3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[3,2,1,4,5] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,1,2,3,5] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,2,1,3,5] => [2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,3,1,2,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,3,2,1,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,2,1,3,4] => [1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,3,1,2,4] => [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,3,2,1,4] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,1,2,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,2,1,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,1,3,4,5,6] => [5,6,4,3,2,1] => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[3,1,2,4,5,6] => [4,6,5,3,2,1] => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[3,2,1,4,5,6] => [4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,1,2,3,5,6] => [3,6,5,4,2,1] => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,2,1,3,5,6] => [3,5,6,4,2,1] => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,3,1,2,5,6] => [3,4,6,5,2,1] => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,3,2,1,5,6] => [3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,1,2,3,4,6] => [2,6,5,4,3,1] => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,2,1,3,4,6] => [2,5,6,4,3,1] => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,3,1,2,4,6] => [2,4,6,5,3,1] => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,3,2,1,4,6] => [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,4,1,2,3,6] => [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,4,2,1,3,6] => [2,3,5,6,4,1] => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,4,3,1,2,6] => [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,4,3,2,1,6] => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,2,1,3,4,5] => [1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,3,1,2,4,5] => [1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,3,2,1,4,5] => [1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,4,1,2,3,5] => [1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,4,2,1,3,5] => [1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.