- FindStat

Your data matches 133 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000264
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,2,5,3] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3,5,4] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[2,4,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,5,1,4,3] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[2,5,3,1,4] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[3,1,4,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[3,1,4,5,2] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,2,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,2,5,1,4] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[4,1,3,5,2] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[4,2,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,5,2,3,6] => [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] => [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,4,5,3,6,2] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,5,1,2,3,6] => ([(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] => [4,5,1,2,3,6] => ([(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] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
[1,4,6,3,5,2] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,4,6,5,2,3] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4

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

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 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

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

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 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

Description

The number of parts from which one can substract 2 and still get an integer partition.

Matching statistic: St000934

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 1 = 4 - 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

Description

The 2-degree of an integer partition. For an integer partition

$\lambda$ , this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by

$\lambda$ .

Matching statistic: St001217

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 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

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

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 4 - 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

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

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 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
[1,2,3,7,4,6,5] => [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

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001594: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 4 - 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
[1,2,3,7,4,6,5] => [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

Mapping

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%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => ? = 4 - 3
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => ? = 4 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 3
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => ? = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 3
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 3
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 4 - 3
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => ? = 4 - 3
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => ? = 4 - 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => ? = 4 - 3
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => ? = 4 - 3
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => ? = 4 - 3
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => ? = 4 - 3
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => ? = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => ? = 4 - 3
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => ? = 4 - 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
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [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
[1,5,6,2,4,3] => [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
[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
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,2,5,7,6,4] => [1,3,2,5,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,2,6,5,4,7] => [1,3,2,6,4,7,5] => [4,3]
 => [3]
 => 1 = 4 - 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
[1,3,5,2,4,7,6] => [1,3,5,2,4,7,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,5,2,7,4,6] => [1,3,5,2,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,5,2,7,6,4] => [1,3,5,2,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,5,4,2,7,6] => [1,3,5,2,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,5,4,6,2,7] => [1,3,5,2,7,4,6] => [4,3]
 => [3]
 => 1 = 4 - 3
[1,3,5,4,7,6,2] => [1,3,5,2,4,7,6] => [4,3]
 => [3]
 => 1 = 4 - 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
[1,3,6,2,4,7,5] => [1,3,6,2,4,7,5] => [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

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)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%

Values

[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 4 - 4
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 4 - 4
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 4
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 4
[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
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 4 - 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
[2,4,1,3,5] => [1,3,5,2,4] => [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
[2,5,1,4,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 4
[2,5,3,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 4
[3,1,4,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 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
[3,2,5,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 4
[4,1,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 4 - 4
[4,2,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 4 - 4
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 4
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 4
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 4
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 4
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 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
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 4
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 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
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 4
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 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
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(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
[2,1,3,4,6,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 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
[2,3,1,4,5,6] => [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
[2,3,6,1,4,5] => [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
[2,4,6,3,5,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 4
[2,4,6,5,3,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 4 - 4
[2,5,1,3,4,6] => [1,3,4,6,2,5] => [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 123 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. 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. St001330The hat guessing number of a graph. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000527The width of the poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000896The number of zeros on the main diagonal of an alternating sign matrix. 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. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ 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. St000068The number of minimal elements in a poset. St001964The interval resolution global dimension of a poset. St001645The pebbling number of a connected graph. St000759The smallest missing part in an integer partition. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000834The number of right outer peaks 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. 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. 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. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001557The number of inversions of the second entry of a permutation. St001668The number of points of the poset minus the width of the poset. St001569The maximal modular displacement of a permutation. St001864The number of excedances of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St000222The number of alignments in the permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001535The number of cyclic alignments of a permutation. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001863The number of weak excedances of a signed permutation. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000702The number of weak deficiencies of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000991The number of right-to-left minima of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001667The maximal size of a pair of weak twins for a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000177The number of free tiles in the pattern. St000197The number of entries equal to positive one in the alternating sign matrix. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000353The number of inner valleys of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000635The number of strictly order preserving maps of a poset into itself. St000710The number of big deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001469The holeyness of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001665The number of pure excedances of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001896The number of right descents of a signed permutations. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St000625The sum of the minimal distances to a greater element. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001616The number of neutral elements in a lattice. St001833The number of linear intervals in a lattice. St001679The number of subsets of a lattice whose meet is the bottom element.