- FindStat

Your data matches 287 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000008

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The major index of the composition. The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St000081

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of edges of a graph.

Matching statistic: St000171

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the graph. This is the maximal vertex degree of a graph.

Matching statistic: St000312

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.

Matching statistic: St000422
(load all 3 compositions to match this statistic)

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph

$K_n$ equals

$2n-2$ . For this reason, we do not define the energy of the empty graph.

Matching statistic: St000987

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

Matching statistic: St001161

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The major index north count of a Dyck path. The descent set

$\operatorname{des}(D)$ of a Dyck path

$D = D_1 \cdots D_{2n}$ with

$D_i \in \{N,E\}$ is given by all indices

$i$ such that

$D_i = E$ and

$D_{i+1} = N$ . This is, the positions of the valleys of

$D$ . The '''major index''' of a Dyck path is then the sum of the positions of the valleys,

$\sum_{i \in \operatorname{des}(D)} i$ , see [[St000027]]. The '''major index north count''' is given by

$\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$ .

Matching statistic: St001307

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001307: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of induced stars on four vertices in a graph.

Matching statistic: St001479

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of bridges of a graph. A bridge is an edge whose removal increases the number of connected components of the graph.

Matching statistic: St001826

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001826: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of leaves on a vertex of a graph.

The following 277 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000468The Hosoya index of a graph. St000763The sum of the positions of the strong records of an integer composition. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000169The cocharge of a standard tableau. St000185The weighted size of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001696The natural major index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St000626The minimal period of a binary word. St000738The first entry in the last row of a standard tableau. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000827The decimal representation of a binary word with a leading 1. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000947The major index east count of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000455The second largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000264The girth of a graph, which is not a tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000667The greatest common divisor of the parts of the partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001763The Hurwitz number of an integer partition. St000145The Dyson rank of a partition. St000464The Schultz index of a connected graph. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001378The product of the cohook lengths of the integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000137The Grundy value of an integer partition. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000997The even-odd crank of an integer partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000311The number of vertices of odd degree in a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001875The number of simple modules with projective dimension at most 1. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000462The major index minus the number of excedences of a permutation. St001703The villainy of a graph. St000154The sum of the descent bottoms of a permutation. St000068The number of minimal elements in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001867The number of alignments of type EN of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001371The length of the longest Yamanouchi prefix of a binary word. St001433The flag major index of a signed permutation. St001524The degree of symmetry of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001819The flag Denert index of a signed permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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

$K[x]/(x^n)$ . St001569The maximal modular displacement of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001768The number of reduced words of a signed permutation. St001853The size of the two-sided Kazhdan-Lusztig cell, St001720The minimal length of a chain of small intervals in a lattice. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001854The size of the left Kazhdan-Lusztig cell, St001845The number of join irreducibles minus the rank of a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St001570The minimal number of edges to add to make a graph Hamiltonian.