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Matching statistic: St001486
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Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[2,1] => [1,1] => [1,1] => 2
[1,3,2] => [2,1] => [2,1] => 3
[2,3,1] => [2,1] => [2,1] => 3
[3,2,1] => [1,1,1] => [1,1,1] => 2
[1,2,4,3] => [3,1] => [3,1] => 3
[1,3,4,2] => [3,1] => [3,1] => 3
[1,4,3,2] => [2,1,1] => [1,2,1] => 4
[2,1,4,3] => [1,2,1] => [2,1,1] => 3
[2,3,4,1] => [3,1] => [3,1] => 3
[2,4,3,1] => [2,1,1] => [1,2,1] => 4
[3,1,4,2] => [1,2,1] => [2,1,1] => 3
[3,2,4,1] => [1,2,1] => [2,1,1] => 3
[3,4,2,1] => [2,1,1] => [1,2,1] => 4
[4,1,3,2] => [1,2,1] => [2,1,1] => 3
[4,2,3,1] => [1,2,1] => [2,1,1] => 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 2
[1,2,3,5,4] => [4,1] => [4,1] => 3
[1,2,4,5,3] => [4,1] => [4,1] => 3
[1,2,5,4,3] => [3,1,1] => [1,3,1] => 4
[1,3,2,5,4] => [2,2,1] => [2,2,1] => 5
[1,3,4,5,2] => [4,1] => [4,1] => 3
[1,3,5,4,2] => [3,1,1] => [1,3,1] => 4
[1,4,2,5,3] => [2,2,1] => [2,2,1] => 5
[1,4,3,5,2] => [2,2,1] => [2,2,1] => 5
[1,4,5,3,2] => [3,1,1] => [1,3,1] => 4
[1,5,2,4,3] => [2,2,1] => [2,2,1] => 5
[1,5,3,4,2] => [2,2,1] => [2,2,1] => 5
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => 4
[2,1,3,5,4] => [1,3,1] => [3,1,1] => 3
[2,1,4,5,3] => [1,3,1] => [3,1,1] => 3
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => 4
[2,3,1,5,4] => [2,2,1] => [2,2,1] => 5
[2,3,4,5,1] => [4,1] => [4,1] => 3
[2,3,5,4,1] => [3,1,1] => [1,3,1] => 4
[2,4,1,5,3] => [2,2,1] => [2,2,1] => 5
[2,4,3,5,1] => [2,2,1] => [2,2,1] => 5
[2,4,5,3,1] => [3,1,1] => [1,3,1] => 4
[2,5,1,4,3] => [2,2,1] => [2,2,1] => 5
[2,5,3,4,1] => [2,2,1] => [2,2,1] => 5
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => 4
[3,1,2,5,4] => [1,3,1] => [3,1,1] => 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => 3
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => 4
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => 3
[3,2,4,5,1] => [1,3,1] => [3,1,1] => 3
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => 4
[3,4,1,5,2] => [2,2,1] => [2,2,1] => 5
[3,4,2,5,1] => [2,2,1] => [2,2,1] => 5
[3,4,5,2,1] => [3,1,1] => [1,3,1] => 4
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000777
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ 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] => ([(0,1)],2)
=> 2
[1,3,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,3,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,4,3] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,4,5,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,3,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,1,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3,4,5,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,5,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,4,1,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,3,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,5,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,5,1,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,3,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,1,2,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,4,5,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,4,1,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,2,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,5,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001035
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1 - 2
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 0 = 2 - 2
[1,3,2] => [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,3,1] => [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,4,3] => [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,3,4,2] => [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,4,3,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[2,1,4,3] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,3,4,1] => [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[2,4,3,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[3,1,4,2] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,4,1] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,4,2,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[4,1,3,2] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[4,2,3,1] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,3,5,4] => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,2,4,5,3] => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,2,5,4,3] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,3,2,5,4] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,3,4,5,2] => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,3,5,4,2] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,4,2,5,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,4,3,5,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,4,5,3,2] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,5,2,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,5,3,4,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[2,1,3,5,4] => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,4,5,3] => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[2,3,1,5,4] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,3,4,5,1] => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[2,3,5,4,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[2,4,1,5,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,4,3,5,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,4,5,3,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[2,5,1,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,5,3,4,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[3,1,2,5,4] => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,1,4,5,2] => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,4,5,1] => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[3,4,1,5,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,4,2,5,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,4,5,2,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[3,5,1,4,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000453
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2
[1,3,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,3,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,4,3] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,4,5,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,3,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,1,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3,4,5,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,5,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,4,1,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,3,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,5,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,5,1,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,3,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,1,2,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,4,5,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,4,1,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,2,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,5,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,1,5,7,6,4,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,1,6,7,5,4,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,2,7,6,5,4] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,4,7,6,5,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,5,7,6,4,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,1,6,7,5,4,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,2,4,7,6,5,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,2,5,7,6,4,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[3,2,6,7,5,4,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,1,2,7,6,5,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,1,3,7,6,5,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,1,5,7,6,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,1,6,7,5,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,2,3,7,6,5,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,2,5,7,6,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,2,6,7,5,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,3,5,7,6,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[4,3,6,7,5,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,1,2,7,6,4,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,1,3,7,6,4,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,1,4,7,6,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,1,6,7,4,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,2,3,7,6,4,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,2,4,7,6,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,2,6,7,4,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,3,4,7,6,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,3,6,7,4,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[5,4,6,7,3,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,1,2,7,5,4,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,1,3,7,5,4,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,1,4,7,5,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,1,5,7,4,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,2,3,7,5,4,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,2,4,7,5,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,2,5,7,4,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,3,4,7,5,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,3,5,7,4,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[6,4,5,7,3,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,1,2,6,5,4,3] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,1,3,6,5,4,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,1,4,6,5,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,1,5,6,4,3,2] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,2,3,6,5,4,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,2,4,6,5,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,2,5,6,4,3,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,3,4,6,5,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,3,5,6,4,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[7,4,5,6,3,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 29%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 - 3
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
=> -1 = 2 - 3
[1,3,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,3,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
[1,2,4,3] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,3,4,2] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,4,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[2,1,4,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,3,4,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,4,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[3,1,4,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,2,4,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,4,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[4,1,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[4,2,3,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
[1,2,3,5,4] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,2,4,5,3] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,2,5,4,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,3,2,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,3,4,5,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,3,5,4,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,4,2,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,4,3,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,4,5,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,5,2,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,5,3,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,1,3,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,1,4,5,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,3,1,5,4] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,3,4,5,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,3,5,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,4,1,5,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,4,3,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,4,5,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,5,1,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,5,3,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[3,1,2,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,2,4,5,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[3,4,1,5,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[3,4,2,5,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[3,4,5,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[3,5,1,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[3,5,2,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[3,5,4,2,1] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[4,1,2,5,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,1,3,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,1,5,3,2] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[4,2,1,5,3] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,2,3,5,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,2,5,3,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[4,3,1,5,2] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,3,2,5,1] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,3,5,2,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[4,5,1,3,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[4,5,2,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[4,5,3,2,1] => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[5,1,2,4,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,1,3,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,1,4,3,2] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[5,2,1,4,3] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,2,3,4,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,2,4,3,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[5,3,1,4,2] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,3,2,4,1] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,3,4,2,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[5,4,1,3,2] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,4,2,3,1] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[5,4,3,2,1] => [1,1,1,1,1] => [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)
=> -1 = 2 - 3
[1,2,3,4,6,5] => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,2,3,5,6,4] => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,2,3,6,5,4] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,2,4,3,6,5] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,2,4,5,6,3] => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,2,4,6,5,3] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,2,5,3,6,4] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,2,5,4,6,3] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,2,5,6,4,3] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,2,6,3,5,4] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,2,6,4,5,3] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,3,2,4,6,5] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,3,2,5,6,4] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,3,4,5,6,2] => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,1,3,4,6,5] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,1,3,5,6,4] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,1,4,5,6,3] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,4,5,6,1] => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[3,1,2,4,6,5] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[3,1,2,5,6,4] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[3,1,4,5,6,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[3,2,1,4,6,5] => [1,1,3,1] => [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 = 3 - 3
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001488
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [[1],[]]
=> 1
[2,1] => [1,1] => [1,1] => [[1,1],[]]
=> 2
[1,3,2] => [2,1] => [2,1] => [[2,2],[1]]
=> 3
[2,3,1] => [2,1] => [2,1] => [[2,2],[1]]
=> 3
[3,2,1] => [1,1,1] => [1,1,1] => [[1,1,1],[]]
=> 2
[1,2,4,3] => [3,1] => [3,1] => [[3,3],[2]]
=> 3
[1,3,4,2] => [3,1] => [3,1] => [[3,3],[2]]
=> 3
[1,4,3,2] => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 4
[2,1,4,3] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 3
[2,3,4,1] => [3,1] => [3,1] => [[3,3],[2]]
=> 3
[2,4,3,1] => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 4
[3,1,4,2] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 3
[3,2,4,1] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 3
[3,4,2,1] => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 4
[4,1,3,2] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 3
[4,2,3,1] => [1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> 2
[1,2,3,5,4] => [4,1] => [4,1] => [[4,4],[3]]
=> 3
[1,2,4,5,3] => [4,1] => [4,1] => [[4,4],[3]]
=> 3
[1,2,5,4,3] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,3,2,5,4] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[1,3,4,5,2] => [4,1] => [4,1] => [[4,4],[3]]
=> 3
[1,3,5,4,2] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,4,2,5,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[1,4,3,5,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[1,4,5,3,2] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,5,2,4,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[1,5,3,4,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[1,5,4,3,2] => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 4
[2,1,3,5,4] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[2,1,4,5,3] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[2,1,5,4,3] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 4
[2,3,1,5,4] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[2,3,4,5,1] => [4,1] => [4,1] => [[4,4],[3]]
=> 3
[2,3,5,4,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[2,4,1,5,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[2,4,3,5,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[2,4,5,3,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[2,5,1,4,3] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[2,5,3,4,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[2,5,4,3,1] => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 4
[3,1,2,5,4] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[3,1,5,4,2] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 4
[3,2,1,5,4] => [1,1,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3
[3,2,4,5,1] => [1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[3,2,5,4,1] => [1,2,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 4
[3,4,1,5,2] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[3,4,2,5,1] => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 5
[3,4,5,2,1] => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,2,3,4,6,5] => [5,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,2,3,5,6,4] => [5,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,2,3,6,5,4] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,2,4,3,6,5] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,2,4,5,6,3] => [5,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,2,4,6,5,3] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,2,5,3,6,4] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,2,5,4,6,3] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,2,5,6,4,3] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,2,6,3,5,4] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,2,6,4,5,3] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 4
[1,3,2,4,6,5] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,3,2,5,6,4] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,3,2,6,5,4] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,3,4,2,6,5] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,3,4,5,6,2] => [5,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,3,4,6,5,2] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,3,5,2,6,4] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,3,5,4,6,2] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,3,5,6,4,2] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,3,6,2,5,4] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,3,6,4,5,2] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 4
[1,4,2,3,6,5] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,4,2,5,6,3] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,4,2,6,5,3] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,4,3,2,6,5] => [2,1,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,4,3,5,6,2] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,4,3,6,5,2] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,4,5,2,6,3] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,4,5,3,6,2] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,4,5,6,3,2] => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,4,6,2,5,3] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,4,6,3,5,2] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,4,6,5,3,2] => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 4
[1,5,2,3,6,4] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,5,2,4,6,3] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,5,2,6,4,3] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,5,3,2,6,4] => [2,1,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,5,3,4,6,2] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,5,3,6,4,2] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,5,4,2,6,3] => [2,1,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,5,4,3,6,2] => [2,1,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,5,4,6,3,2] => [2,2,1,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 5
[1,5,6,2,4,3] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,5,6,3,4,2] => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,5,6,4,3,2] => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 4
[1,6,2,3,5,4] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,6,2,4,5,3] => [2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
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