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Matching statistic: St000863
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St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 4
[2,1,4,3] => 3
[2,3,1,4] => 4
[2,3,4,1] => 4
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 3
Description
The length of the first row of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the length of the first row of $P$ and $Q$.
Matching statistic: St001880
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 1
[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] => [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[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] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[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] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[1,4,5,3,2] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,5,2,3,4] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,5,2,4,3] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[1,5,3,2,4] => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[1,5,3,4,2] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3
[1,5,4,2,3] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[1,5,4,3,2] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,1,3,4,5] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> ([(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)
=> ? = 4
[2,1,4,3,5] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[2,1,4,5,3] => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[2,1,5,3,4] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,1,5,4,3] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[2,3,1,4,5] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,1,5,4] => [4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[2,3,4,1,5] => [5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,4,5,1] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,1,4,5] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,4,1,5] => [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,4,5,1] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,2,1,5] => [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,2,5,1] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,2,1,5] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,2,5,1] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,5,2,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [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] => [6,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,1,4,5,6] => [6,5,4,1,3,2] => [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,4,1,5,6] => [6,5,1,4,3,2] => [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,4,5,1,6] => [6,1,5,4,3,2] => [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,4,5,6,1] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,2,1,4,5,6] => [6,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,2,4,1,5,6] => [6,5,1,4,2,3] => [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,2,4,5,1,6] => [6,1,5,4,2,3] => [[.,[[[.,[.,.]],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,2,4,5,6,1] => [1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,2,1,5,6] => [6,5,1,2,4,3] => [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,2,5,1,6] => [6,1,5,2,4,3] => [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,2,5,6,1] => [1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,2,1,6] => [6,1,2,5,4,3] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,2,6,1] => [1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,6,2,1] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,2,1,5,6] => [6,5,1,2,3,4] => [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,2,5,1,6] => [6,1,5,2,3,4] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,2,5,6,1] => [1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,5,2,1,6] => [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,5,2,6,1] => [1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,3,5,6,2,1] => [1,2,6,5,3,4] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(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
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 1 - 1
[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] => [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2 - 1
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2 - 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] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(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)
=> ? = 3 - 1
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 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] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,5,2,3,4] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,5,3,2,4] => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[1,5,3,4,2] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 - 1
[1,5,4,2,3] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,5,4,3,2] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,1,3,4,5] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> ([(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)
=> ? = 4 - 1
[2,1,4,3,5] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[2,1,4,5,3] => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,1,5,3,4] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,1,5,4,3] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[2,3,1,4,5] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,1,5,4] => [4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[2,3,4,1,5] => [5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,4,5,1] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,1,4,5] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,4,1,5] => [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,2,1,5] => [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,2,5,1] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,2,1,5] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,2,5,1] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,2,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [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] => [6,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,1,4,5,6] => [6,5,4,1,3,2] => [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,4,1,5,6] => [6,5,1,4,3,2] => [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,4,5,1,6] => [6,1,5,4,3,2] => [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,3,4,5,6,1] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,2,1,4,5,6] => [6,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,2,4,1,5,6] => [6,5,1,4,2,3] => [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,2,4,5,1,6] => [6,1,5,4,2,3] => [[.,[[[.,[.,.]],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,2,4,5,6,1] => [1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,2,1,5,6] => [6,5,1,2,4,3] => [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,2,5,1,6] => [6,1,5,2,4,3] => [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,2,5,6,1] => [1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,2,1,6] => [6,1,2,5,4,3] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,2,6,1] => [1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,2,1,5,6] => [6,5,1,2,3,4] => [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,2,5,1,6] => [6,1,5,2,3,4] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,2,5,6,1] => [1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,5,2,1,6] => [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,5,2,6,1] => [1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,3,5,6,2,1] => [1,2,6,5,3,4] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(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.
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