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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000264
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Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,6,4,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,2,6,3] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,5,3,2,6] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,6,2,3] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,5,6,3,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,4,6,2,3,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,6,5,2,3] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,4,6,5,3,2] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,2,6,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,3,4,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,4,3,2,6] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[1,5,6,2,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,5,6,4,2,3] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,5,6,4,3,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000149
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000256
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000934
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> 0 = 3 - 3
Description
The 2-degree of an integer partition.
For an integer partition λ, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by λ.
Matching statistic: St001217
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St001292
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 3 - 3
Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Here A is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].
Matching statistic: St001184
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,4,6,5,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,5,3,4,6] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,2,7,6,4,3,5] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,2,4,6,3,7] => [1,5,2,4,6,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,2,4,6,7,3] => [1,5,2,4,6,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,3,2,4,6,7] => [1,5,2,4,6,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,3,6,7,2,4] => [1,5,2,4,3,6,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,5,3,7,2,4,6] => [1,5,2,4,6,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,3,5,7,4] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,4,3,5,7] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,4,5,3,7] => [1,6,2,4,5,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,4,5,7,3] => [1,6,2,4,5,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,5,3,4,7] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,2,5,4,7,3] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,2,4,5,7] => [1,6,2,4,5,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,2,5,4,7] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,4,7,2,5] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,5,4,7,2] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,5,7,2,4] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,5,7,4,2] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,3,7,2,4,5] => [1,6,2,4,5,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,2,3,5,7] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,3,5,7,2] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,7,2,3,5] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,7,2,5,3] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,7,3,2,5] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,4,7,3,5,2] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,5,7,2,4,3] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,6,5,7,3,2,4] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001594
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001594: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001594: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 3 - 3
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,4,6,5,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,5,3,4,6] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,2,7,6,4,3,5] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,2,4,6,3,7] => [1,5,2,4,6,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,2,4,6,7,3] => [1,5,2,4,6,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,3,2,4,6,7] => [1,5,2,4,6,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,3,6,7,2,4] => [1,5,2,4,3,6,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,5,3,7,2,4,6] => [1,5,2,4,6,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,3,5,7,4] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,4,3,5,7] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,4,5,3,7] => [1,6,2,4,5,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,4,5,7,3] => [1,6,2,4,5,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,5,3,4,7] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,2,5,4,7,3] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,2,4,5,7] => [1,6,2,4,5,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,2,5,4,7] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,4,7,2,5] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,5,4,7,2] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,5,7,2,4] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,5,7,4,2] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,3,7,2,4,5] => [1,6,2,4,5,3,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,2,3,5,7] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,3,5,7,2] => [1,6,2,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,7,2,3,5] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,7,2,5,3] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,7,3,2,5] => [1,6,2,5,3,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,4,7,3,5,2] => [1,6,2,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,5,7,2,4,3] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
[1,6,5,7,3,2,4] => [1,6,2,4,3,5,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 3
Description
The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied.
See the link for the definition.
Matching statistic: St001604
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 4 - 3
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 3 - 3
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> ? = 4 - 3
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [4,2]
=> [2]
=> ? = 4 - 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [4,1,1]
=> [1,1]
=> ? = 3 - 3
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[2,5,1,4,6,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[2,5,3,1,4,6] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[2,5,4,1,3,6] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,1,4,6,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,1,5,6,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,4,1,5,6] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,5,1,4,6] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,1,3,6,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,2,5,1,3,6] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,2,6,7,4,5] => [1,3,2,6,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,2,6,7,5,4] => [1,3,2,6,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,2,7,5,4,6] => [1,3,2,7,4,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,5,7,2,4,6] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,5,7,2,6,4] => [1,3,5,7,2,6,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,5,7,4,2,6] => [1,3,5,7,2,6,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,5,7,4,6,2] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,5,7,6,2,4] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,5,7,6,4,2] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,2,5,4,7] => [1,3,6,2,5,4,7] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,2,5,7,4] => [1,3,6,2,5,7,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,2,7,4,5] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,2,7,5,4] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,4,2,5,7] => [1,3,6,2,5,7,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,4,2,7,5] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,4,5,2,7] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,4,7,2,5] => [1,3,6,2,5,4,7] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,5,2,7,4] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,5,4,2,7] => [1,3,6,2,7,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,7,2,4,5] => [1,3,6,7,2,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
[1,3,6,7,2,5,4] => [1,3,6,7,2,5,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,7,4,2,5] => [1,3,6,7,2,5,4] => [4,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,3,6,7,4,5,2] => [1,3,6,7,2,4,5] => [4,3]
=> [3]
=> 1 = 4 - 3
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[3,1,4,5,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[3,1,5,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[3,2,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[3,2,4,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,6,2,3,5] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,4,6,5,3,2] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,3,2,4,6] => [1,5,2,4,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,3,2,4,5] => [1,6,2,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,3,5,4,2] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,4,2,3,5] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,4,3,5,2] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,1,3,5,6,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,4,5,3,6] => [1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,4,5,6,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,4,6,3,5] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,4,6,5,3] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,5,6,3,4] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,5,6,4,3] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,1,6,3,5,4] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,1,6,4,3,5] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,3,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,1,4,6,5] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,1,5,6,4] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,4,1,5,6] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,5,1,4,6] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,5,1,6,4] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,3,5,4,1,6] => [1,6,2,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,3,6,1,4,5] => [1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,4,1,3,5,6] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
[2,5,6,3,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,5,6,4,3,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,1,2,5,6,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,1,4,5,6,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,1,4,6,5,2] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,1,5,6,4,2] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,2,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,2,1,4,6,5] => [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,2,1,5,6,4] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,2,5,6,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,4,1,2,5,6] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,4,1,5,6,2] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,4,2,1,5,6] => [1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,4,2,5,6,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
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.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000654The first descent of a permutation. St000917The open packing number of a graph. St000990The first ascent of a permutation. St001672The restrained domination number of a graph. St001737The number of descents of type 2 in a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001826The maximal number of leaves on a vertex of a graph. St001828The Euler characteristic of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St000527The width of the poset. St001618The cardinality of the Frattini sublattice of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000909The number of maximal chains of maximal size in a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000754The Grundy value for the game of removing nestings in a perfect matching. St001964The interval resolution global dimension of a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000068The number of minimal elements in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000546The number of global descents of a permutation. St001115The number of even descents of a permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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