Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ 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
[4,1,2,3] => [1,4,3,2] => [1,2,1] => 4
[4,2,1,3] => [1,4,3,2] => [1,2,1] => 4
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => 5
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => 5
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => 5
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => 5
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => 4
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => 4
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => 4
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => 4
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => 4
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => 4
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => 4
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => 4
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => 4
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => 4
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => 4
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => 4
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => 5
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => 5
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => 5
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => 5
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => 5
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => 5
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => 5
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => 5
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => 5
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => 5
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => 5
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => 5
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => 5
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => 5
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => 5
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => 5
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => 5
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => 5
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => 5
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => 5
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => 5
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => 5
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => 5
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => 5
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
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ 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
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000453
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
 => 1
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [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)
 => 5
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[6,1,2,3,7,5,4] => [1,6,5,7,4,3,2] => [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
[6,1,2,7,3,4,5] => [1,6,4,7,5,3,2] => [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
[6,2,1,3,7,5,4] => [1,6,5,7,4,3,2] => [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
[6,2,1,7,3,4,5] => [1,6,4,7,5,3,2] => [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
[7,1,2,3,6,4,5] => [1,7,5,6,4,3,2] => [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
[7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => [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
[7,1,2,6,3,5,4] => [1,7,4,6,5,3,2] => [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
[7,1,5,2,4,3,6] => [1,7,6,3,5,4,2] => [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
[7,2,1,3,6,4,5] => [1,7,5,6,4,3,2] => [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
[7,2,1,5,3,4,6] => [1,7,6,4,5,3,2] => [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
[7,2,1,6,3,5,4] => [1,7,4,6,5,3,2] => [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
[7,2,5,1,4,3,6] => [1,7,6,3,5,4,2] => [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
[7,5,1,3,4,2,6] => [1,7,6,2,5,4,3] => [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
[7,5,2,3,4,1,6] => [1,7,6,2,5,4,3] => [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
[7,5,3,1,4,2,6] => [1,7,6,2,5,4,3] => [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
[7,5,3,2,4,1,6] => [1,7,6,2,5,4,3] => [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: St000894
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[1] => [1,0]
 => [1,0]
 => [[1]]
 => ? = 1 - 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0 = 4 - 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0 = 4 - 4
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1 = 5 - 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1 = 5 - 4
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1 = 5 - 4
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1 = 5 - 4
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0 = 4 - 4
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0 = 4 - 4
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 1 = 5 - 4
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 1 = 5 - 4
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,3,6,4,1,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 1 = 5 - 4
[2,5,3,1,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 1 = 5 - 4
[2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,3,1,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,3,5,1,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,5,3,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5 - 4
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,5,6,1,4,2] => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,5,6,2,4,1] => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,5,6,4,1,2] => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,5,6,4,2,1] => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[3,6,1,4,2,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[3,6,2,1,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[3,6,2,4,1,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[4,1,5,6,2,3] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[4,2,1,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[4,2,5,6,1,3] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[4,2,6,3,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 6 - 4
[4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 4 - 4
[5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,2,4,1,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,1,3,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,3,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
[5,4,3,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 4 - 4
Description
The trace of an alternating sign matrix.
Matching statistic: St000264
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[1] => => [1] => ([],1)
 => ? = 1 - 2
[4,1,2,3] => 000 => [4] => ([],4)
 => ? = 4 - 2
[4,2,1,3] => 000 => [4] => ([],4)
 => ? = 4 - 2
[1,5,2,3,4] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 5 - 2
[1,5,3,2,4] => 1000 => [1,4] => ([(3,4)],5)
 => ? = 5 - 2
[2,5,1,3,4] => 0000 => [5] => ([],5)
 => ? = 5 - 2
[2,5,3,1,4] => 0000 => [5] => ([],5)
 => ? = 5 - 2
[3,1,5,2,4] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[3,2,5,1,4] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[4,1,2,5,3] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[4,2,1,5,3] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,1,2,3,4] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,1,4,2,3] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,2,1,3,4] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,2,4,1,3] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,4,1,3,2] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,4,2,3,1] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,4,3,1,2] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[5,4,3,2,1] => 0000 => [5] => ([],5)
 => ? = 4 - 2
[1,2,6,3,4,5] => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,2,6,4,3,5] => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 5 - 2
[1,3,6,2,4,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,3,6,4,2,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,4,2,6,3,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,4,3,6,2,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,5,2,3,6,4] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,5,3,2,6,4] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,2,3,4,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,2,5,3,4] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,3,2,4,5] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,3,5,2,4] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,5,2,4,3] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,5,3,4,2] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,5,4,2,3] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[1,6,5,4,3,2] => 10000 => [1,5] => ([(4,5)],6)
 => ? = 5 - 2
[2,1,6,3,4,5] => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 5 - 2
[2,1,6,4,3,5] => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 5 - 2
[2,3,6,1,4,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,3,6,4,1,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,4,1,6,3,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,4,3,6,1,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,5,1,3,6,4] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,5,3,1,6,4] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,1,3,4,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,1,5,3,4] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,3,1,4,5] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,3,5,1,4] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,5,1,4,3] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,5,3,4,1] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,5,4,1,3] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[2,6,5,4,3,1] => 00000 => [6] => ([],6)
 => ? = 5 - 2
[3,1,4,6,2,5] => 00000 => [6] => ([],6)
 => ? = 4 - 2
[1,2,3,7,4,5,6] => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,3,7,5,4,6] => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,4,7,3,5,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,4,7,5,3,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,5,3,7,4,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,5,4,7,3,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,6,3,4,7,5] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,6,4,3,7,5] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,3,4,5,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,3,6,4,5] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,4,3,5,6] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,4,6,3,5] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,6,3,5,4] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,6,4,5,3] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,6,5,3,4] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,2,7,6,5,4,3] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,3,2,7,4,5,6] => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[1,3,2,7,5,4,6] => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[2,1,3,7,4,5,6] => 011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
[2,1,3,7,5,4,6] => 011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 5 - 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001198
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1 - 3
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4 - 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4 - 3
[1,5,2,3,4] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 3
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 3
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[4,2,1,5,3] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,1,2,3,4] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,1,4,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,2,1,3,4] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,5,2,3,6,4] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,2,3,4,5] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,2,5,3,4] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,3,4,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,4,2,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[2,1,6,3,4,5] => [2,1,6,4,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,1,6,4,3,5] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,3,6,4,1,5] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 5 - 3
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,5,3,1,6,4] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,3,1,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,1,4,3] => [4,6,5,1,3,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,3,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,4,1,3] => [5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,4,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,1,6,2,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,1,6,5,2,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,4,6,1,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,2,6,5,1,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,1,4,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,2,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,4,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,4,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,6,1,2,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,1,4,2,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,2,1,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,2,4,1,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[4,1,2,5,6,3] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[4,1,2,6,3,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[4,1,5,6,2,3] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[4,1,6,3,2,5] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1] => [1,0]
 => ? = 1 - 3
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4 - 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4 - 3
[1,5,2,3,4] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 3
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 3
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[4,2,1,5,3] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,1,2,3,4] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,1,4,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,2,1,3,4] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 - 3
[1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,5,2,3,6,4] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,2,3,4,5] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,2,5,3,4] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,3,4,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,4,2,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 5 - 3
[2,1,6,3,4,5] => [2,1,6,4,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,1,6,4,3,5] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 5 - 3
[2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,3,6,4,1,5] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 5 - 3
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,5,3,1,6,4] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,3,1,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,1,4,3] => [4,6,5,1,3,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,3,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,4,1,3] => [5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[2,6,5,4,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 - 3
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,1,6,2,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,1,6,5,2,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,4,6,1,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,2,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,2,6,5,1,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,1,4,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,2,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,4,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,5,6,4,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[3,6,1,2,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,1,4,2,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,2,1,4,5] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[3,6,2,4,1,5] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[4,1,2,5,6,3] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[4,1,2,6,3,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
[4,1,5,6,2,3] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 - 3
[4,1,6,3,2,5] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 3
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St001491
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [] => => ? = 1 - 4
[4,1,2,3] => [4,2,3,1] => [2,3,1] => 00 => ? = 4 - 4
[4,2,1,3] => [4,3,2,1] => [3,2,1] => 00 => ? = 4 - 4
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,4,2] => 100 => 1 = 5 - 4
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,2] => 100 => 1 = 5 - 4
[2,5,1,3,4] => [3,5,1,4,2] => [3,1,4,2] => 000 => ? = 5 - 4
[2,5,3,1,4] => [4,5,3,1,2] => [4,3,1,2] => 000 => ? = 5 - 4
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,1,3] => 000 => ? = 4 - 4
[3,2,5,1,4] => [4,2,5,1,3] => [4,2,1,3] => 000 => ? = 4 - 4
[4,1,2,5,3] => [5,2,3,4,1] => [2,3,4,1] => 000 => ? = 4 - 4
[4,2,1,5,3] => [5,3,2,4,1] => [3,2,4,1] => 000 => ? = 4 - 4
[5,1,2,3,4] => [5,2,3,4,1] => [2,3,4,1] => 000 => ? = 4 - 4
[5,1,4,2,3] => [5,2,4,3,1] => [2,4,3,1] => 000 => ? = 4 - 4
[5,2,1,3,4] => [5,3,2,4,1] => [3,2,4,1] => 000 => ? = 4 - 4
[5,2,4,1,3] => [5,4,3,2,1] => [4,3,2,1] => 000 => ? = 4 - 4
[5,4,1,3,2] => [5,4,3,2,1] => [4,3,2,1] => 000 => ? = 4 - 4
[5,4,2,3,1] => [5,4,3,2,1] => [4,3,2,1] => 000 => ? = 4 - 4
[5,4,3,1,2] => [5,4,3,2,1] => [4,3,2,1] => 000 => ? = 4 - 4
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 000 => ? = 4 - 4
[1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,4,5,3] => 1100 => 1 = 5 - 4
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [1,2,5,4,3] => 1100 => 1 = 5 - 4
[1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,2,5,3] => 1000 => 1 = 5 - 4
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [1,5,4,2,3] => 1000 => 1 = 5 - 4
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,5,3,2,4] => 1000 => 1 = 5 - 4
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [1,5,3,2,4] => 1000 => 1 = 5 - 4
[1,5,2,3,6,4] => [1,6,3,4,5,2] => [1,3,4,5,2] => 1000 => 1 = 5 - 4
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [1,4,3,5,2] => 1000 => 1 = 5 - 4
[1,6,2,3,4,5] => [1,6,3,4,5,2] => [1,3,4,5,2] => 1000 => 1 = 5 - 4
[1,6,2,5,3,4] => [1,6,3,5,4,2] => [1,3,5,4,2] => 1000 => 1 = 5 - 4
[1,6,3,2,4,5] => [1,6,4,3,5,2] => [1,4,3,5,2] => 1000 => 1 = 5 - 4
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 5 - 4
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 5 - 4
[1,6,5,3,4,2] => [1,6,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 5 - 4
[1,6,5,4,2,3] => [1,6,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 5 - 4
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 5 - 4
[2,1,6,3,4,5] => [2,1,6,4,5,3] => [2,1,4,5,3] => 0100 => 1 = 5 - 4
[2,1,6,4,3,5] => [2,1,6,5,4,3] => [2,1,5,4,3] => 0100 => 1 = 5 - 4
[2,3,6,1,4,5] => [4,2,6,1,5,3] => [4,2,1,5,3] => 0000 => ? = 5 - 4
[2,3,6,4,1,5] => [5,2,6,4,1,3] => [5,2,4,1,3] => 0000 => ? = 5 - 4
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [3,5,1,2,4] => 0000 => ? = 5 - 4
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [5,3,2,1,4] => 0000 => ? = 5 - 4
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [3,1,4,5,2] => 0000 => ? = 5 - 4
[2,5,3,1,6,4] => [4,6,3,1,5,2] => [4,3,1,5,2] => 0000 => ? = 5 - 4
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [3,1,4,5,2] => 0000 => ? = 5 - 4
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [3,1,5,4,2] => 0000 => ? = 5 - 4
[2,6,3,1,4,5] => [4,6,3,1,5,2] => [4,3,1,5,2] => 0000 => ? = 5 - 4
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [5,3,4,1,2] => 0000 => ? = 5 - 4
[2,6,5,1,4,3] => [4,6,5,1,3,2] => [4,5,1,3,2] => 0000 => ? = 5 - 4
[2,6,5,3,4,1] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 5 - 4
[2,6,5,4,1,3] => [5,6,4,3,1,2] => [5,4,3,1,2] => 0000 => ? = 5 - 4
[2,6,5,4,3,1] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 5 - 4
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [5,2,3,1,4] => 0000 => ? = 4 - 4
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [4,2,1,5,3] => 0000 => ? = 4 - 4
[3,1,6,2,4,5] => [4,2,6,1,5,3] => [4,2,1,5,3] => 0000 => ? = 6 - 4
[3,1,6,5,2,4] => [5,2,6,4,1,3] => [5,2,4,1,3] => 0000 => ? = 4 - 4
[3,2,4,6,1,5] => [5,2,3,6,1,4] => [5,2,3,1,4] => 0000 => ? = 4 - 4
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [4,2,1,5,3] => 0000 => ? = 4 - 4
[3,2,6,1,4,5] => [4,2,6,1,5,3] => [4,2,1,5,3] => 0000 => ? = 6 - 4
[3,2,6,5,1,4] => [5,2,6,4,1,3] => [5,2,4,1,3] => 0000 => ? = 4 - 4
[3,5,6,1,4,2] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 4 - 4
[3,5,6,2,4,1] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 4 - 4
[3,5,6,4,1,2] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 4 - 4
[3,5,6,4,2,1] => [6,5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 4 - 4
[3,6,1,2,4,5] => [4,6,3,1,5,2] => [4,3,1,5,2] => 0000 => ? = 6 - 4
[3,6,1,4,2,5] => [5,6,3,4,1,2] => [5,3,4,1,2] => 0000 => ? = 6 - 4
[3,6,2,1,4,5] => [4,6,3,1,5,2] => [4,3,1,5,2] => 0000 => ? = 6 - 4
[3,6,2,4,1,5] => [5,6,3,4,1,2] => [5,3,4,1,2] => 0000 => ? = 6 - 4
[4,1,2,5,6,3] => [6,2,3,4,5,1] => [2,3,4,5,1] => 0000 => ? = 4 - 4
[4,1,2,6,3,5] => [5,2,3,6,1,4] => [5,2,3,1,4] => 0000 => ? = 6 - 4
[4,1,5,6,2,3] => [6,2,5,4,3,1] => [2,5,4,3,1] => 0000 => ? = 4 - 4
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.