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Your data matches 1036 different statistics following compositions of up to 3 maps.
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Matching statistic: St000824
(load all 105 compositions to match this statistic)
(load all 105 compositions to match this statistic)
St000824: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => 2 = 1 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 2 = 1 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => 2 = 1 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 4 = 3 + 1
[2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => 2 = 1 + 1
[1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => 2 = 1 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 4 = 3 + 1
[1,3,4,2,5] => 2 = 1 + 1
[1,4,2,3,5] => 2 = 1 + 1
[2,1,3,4,5] => 2 = 1 + 1
[2,1,3,5,4] => 4 = 3 + 1
[2,1,4,3,5] => 4 = 3 + 1
[2,3,1,4,5] => 2 = 1 + 1
[3,1,2,4,5] => 2 = 1 + 1
[1,2,3,4,6,5] => 2 = 1 + 1
[1,2,3,5,4,6] => 2 = 1 + 1
[1,2,3,5,6,4] => 2 = 1 + 1
[1,2,4,3,5,6] => 2 = 1 + 1
[1,2,4,3,6,5] => 4 = 3 + 1
[1,2,4,5,3,6] => 2 = 1 + 1
[1,3,2,4,5,6] => 2 = 1 + 1
[1,3,2,4,6,5] => 4 = 3 + 1
[1,3,2,5,4,6] => 4 = 3 + 1
[1,3,4,2,5,6] => 2 = 1 + 1
[2,1,3,4,5,6] => 2 = 1 + 1
[2,1,3,4,6,5] => 4 = 3 + 1
[2,1,3,5,4,6] => 4 = 3 + 1
[2,1,4,3,5,6] => 4 = 3 + 1
[2,3,1,4,5,6] => 2 = 1 + 1
Description
The sum of the number of descents and the number of recoils of a permutation.
This statistic is the sum of [[St000021]] and [[St000354]].
Matching statistic: St001535
(load all 116 compositions to match this statistic)
(load all 116 compositions to match this statistic)
St001535: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 0 = 1 - 1
[3,1,2,4] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[2,1,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => 2 = 3 - 1
[2,1,4,3,5] => 2 = 3 - 1
[2,3,1,4,5] => 0 = 1 - 1
[3,1,2,4,5] => 0 = 1 - 1
[1,2,3,4,6,5] => 0 = 1 - 1
[1,2,3,5,4,6] => 0 = 1 - 1
[1,2,3,5,6,4] => 0 = 1 - 1
[1,2,4,3,5,6] => 0 = 1 - 1
[1,2,4,3,6,5] => 2 = 3 - 1
[1,2,4,5,3,6] => 0 = 1 - 1
[1,3,2,4,5,6] => 0 = 1 - 1
[1,3,2,4,6,5] => 2 = 3 - 1
[1,3,2,5,4,6] => 2 = 3 - 1
[1,3,4,2,5,6] => 0 = 1 - 1
[2,1,3,4,5,6] => 0 = 1 - 1
[2,1,3,4,6,5] => 2 = 3 - 1
[2,1,3,5,4,6] => 2 = 3 - 1
[2,1,4,3,5,6] => 2 = 3 - 1
[2,3,1,4,5,6] => 0 = 1 - 1
Description
The number of cyclic alignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [1]
Matching statistic: St001536
(load all 155 compositions to match this statistic)
(load all 155 compositions to match this statistic)
St001536: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 0 = 1 - 1
[3,1,2,4] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[2,1,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => 2 = 3 - 1
[2,1,4,3,5] => 2 = 3 - 1
[2,3,1,4,5] => 0 = 1 - 1
[3,1,2,4,5] => 0 = 1 - 1
[1,2,3,4,6,5] => 0 = 1 - 1
[1,2,3,5,4,6] => 0 = 1 - 1
[1,2,3,5,6,4] => 0 = 1 - 1
[1,2,4,3,5,6] => 0 = 1 - 1
[1,2,4,3,6,5] => 2 = 3 - 1
[1,2,4,5,3,6] => 0 = 1 - 1
[1,3,2,4,5,6] => 0 = 1 - 1
[1,3,2,4,6,5] => 2 = 3 - 1
[1,3,2,5,4,6] => 2 = 3 - 1
[1,3,4,2,5,6] => 0 = 1 - 1
[2,1,3,4,5,6] => 0 = 1 - 1
[2,1,3,4,6,5] => 2 = 3 - 1
[2,1,3,5,4,6] => 2 = 3 - 1
[2,1,4,3,5,6] => 2 = 3 - 1
[2,3,1,4,5,6] => 0 = 1 - 1
Description
The number of cyclic misalignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, \pi(i), j, \pi(j)$ are cyclically ordered and all distinct, see Section 5 of [1].
Matching statistic: St000071
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00065: Permutations āpermutation posetā¶ Posets
St000071: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000071: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => ([],2)
=> 2 = 1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 4 = 3 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4 = 3 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2 = 1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2 = 1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> 4 = 3 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 2 = 1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> 4 = 3 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 4 = 3 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 4 = 3 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> 4 = 3 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> 4 = 3 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000311
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000311: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000311: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 3 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The number of vertices of odd degree in a graph.
Matching statistic: St000312
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000312: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000312: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 3 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4 = 3 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The number of leaves in a graph.
That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000673
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00159: Permutations āDemazure product with inverseā¶ Permutations
St000673: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000673: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,2,1] => 2 = 1 + 1
[3,1,2] => [3,2,1] => 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => 4 = 3 + 1
[2,3,1,4] => [3,2,1,4] => 2 = 1 + 1
[3,1,2,4] => [3,2,1,4] => 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 4 = 3 + 1
[1,3,4,2,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => 4 = 3 + 1
[2,1,4,3,5] => [2,1,4,3,5] => 4 = 3 + 1
[2,3,1,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[3,1,2,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => 2 = 1 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => 2 = 1 + 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => 2 = 1 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => 2 = 1 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => 4 = 3 + 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => 2 = 1 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => 2 = 1 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => 4 = 3 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => 4 = 3 + 1
[1,3,4,2,5,6] => [1,4,3,2,5,6] => 2 = 1 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => 2 = 1 + 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => 4 = 3 + 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => 4 = 3 + 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => 4 = 3 + 1
[2,3,1,4,5,6] => [3,2,1,4,5,6] => 2 = 1 + 1
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St000707
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations āLLPSā¶ Integer partitions
St000707: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000707: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 4 = 3 + 1
[2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> 4 = 3 + 1
[2,1,4,3,5] => [2,2,1]
=> 4 = 3 + 1
[2,3,1,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,6,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,2,4,5,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,4,2,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,3,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,4,3,5,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,3,1,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations āLLPSā¶ Integer partitions
St000708: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000708: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 4 = 3 + 1
[2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> 4 = 3 + 1
[2,1,4,3,5] => [2,2,1]
=> 4 = 3 + 1
[2,3,1,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,6,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,2,4,5,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,4,2,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,3,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,4,3,5,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,3,1,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations āLLPSā¶ Integer partitions
St000933: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000933: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 4 = 3 + 1
[2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> 4 = 3 + 1
[2,1,4,3,5] => [2,2,1]
=> 4 = 3 + 1
[2,3,1,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,6,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,2,4,5,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[1,3,4,2,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,3,4,6,5] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,3,5,4,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,1,4,3,5,6] => [2,2,1,1]
=> 4 = 3 + 1
[2,3,1,4,5,6] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
The following 1026 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001458The rank of the adjacency matrix of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001902The number of potential covers of a poset. St000037The sign of a permutation. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000025The number of initial rises of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000185The weighted size of a partition. St000472The sum of the ascent bottoms of a permutation. St000632The jump number of the poset. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000984The number of boxes below precisely one peak. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001564The value of the forgotten symmetric functions when all variables set to 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001874Lusztig's a-function for the symmetric group. St001961The sum of the greatest common divisors of all pairs of parts. St000039The number of crossings of a permutation. St000068The number of minimal elements in a poset. St000143The largest repeated part of a partition. St000235The number of indices that are not cyclical small weak excedances. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000343The number of spanning subgraphs of a graph. St000346The number of coarsenings of a partition. St000350The sum of the vertex degrees of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000422The energy of a graph, if it is integral. St000439The position of the first down step of a Dyck path. St000465The first Zagreb index of a graph. St000468The Hosoya index of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000527The width of the poset. St000571The F-index (or forgotten topological index) of a graph. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000912The number of maximal antichains in a poset. St000938The number of zeros of the symmetric group character corresponding to the partition. St000972The composition number of a graph. St000995The largest even part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001268The size of the largest ordinal summand in the poset. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001346The number of parking functions that give the same permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001399The distinguishing number of a poset. St001459The number of zero columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001500The global dimension of magnitude 1 Nakayama algebras. St001525The number of symmetric hooks on the diagonal of a partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001779The order of promotion on the set of linear extensions of a poset. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001939The number of parts that are equal to their multiplicity in the integer partition. St000146The Andrews-Garvan crank of a partition. St000187The determinant of an alternating sign matrix. St000722The number of different neighbourhoods in a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000656The number of cuts of a poset. St000001The number of reduced words for a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000041The number of nestings of a perfect matching. St000047The number of standard immaculate tableaux of a given shape. St000054The first entry of the permutation. St000055The inversion sum of a permutation. St000057The Shynar inversion number of a standard tableau. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000081The number of edges of a graph. St000096The number of spanning trees of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000133The "bounce" of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000172The Grundy number of a graph. St000182The number of permutations whose cycle type is the given integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000267The number of maximal spanning forests contained in a graph. St000271The chromatic index of a graph. St000297The number of leading ones in a binary word. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000330The (standard) major index of a standard tableau. St000332The positive inversions of an alternating sign matrix. St000335The difference of lower and upper interactions. St000336The leg major index of a standard tableau. St000345The number of refinements of a partition. St000347The inversion sum of a binary word. St000354The number of recoils of a permutation. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000391The sum of the positions of the ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000446The disorder of a permutation. St000456The monochromatic index of a connected graph. St000490The intertwining number of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000542The number of left-to-right-minima of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000670The reversal length of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000795The mad of a permutation. St000797The stat`` of a permutation. St000809The reduced reflection length of the permutation. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000935The number of ordered refinements of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000971The smallest closer of a set partition. St000976The sum of the positions of double up-steps of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000988The orbit size of a permutation under Foata's bijection. St000991The number of right-to-left minima of a permutation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001108The 2-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001120The length of a longest path in a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001303The number of dominating sets of vertices of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001375The pancake length of a permutation. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001494The Alon-Tarsi number of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001546The number of monomials in the Tutte polynomial of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001649The length of a longest trail in a graph. St001670The connected partition number of a graph. St001694The number of maximal dissociation sets in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St001726The number of visible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001827The number of two-component spanning forests of a graph. St001833The number of linear intervals in a lattice. St001850The number of Hecke atoms of a permutation. St001869The maximum cut size of a graph. St001930The weak major index of a binary word. St001931The weak major index of an integer composition regarded as a word. St001963The tree-depth of a graph. St000008The major index of the composition. St000010The length of the partition. St000027The major index of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000058The order of a permutation. St000069The number of maximal elements of a poset. St000082The number of elements smaller than a binary tree in Tamari order. St000086The number of subgraphs. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000142The number of even parts of a partition. St000217The number of occurrences of the pattern 312 in a permutation. St000223The number of nestings in the permutation. St000268The number of strongly connected orientations of a graph. St000272The treewidth of a graph. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000313The number of degree 2 vertices of a graph. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000389The number of runs of ones of odd length in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000444The length of the maximal rise of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000485The length of the longest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000507The number of ascents of a standard tableau. St000536The pathwidth of a graph. St000538The number of even inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000552The number of cut vertices of a graph. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. 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. St000825The sum of the major and the inverse major index of a permutation. St000828The spearman's rho of a permutation and the identity permutation. St000830The total displacement of a permutation. St000836The number of descents of distance 2 of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000883The number of longest increasing subsequences of a permutation. St000915The Ore degree of a graph. St000926The clique-coclique number of a graph. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000979Half of MacMahon's equal index of a Dyck path. St000989The number of final rises of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001083The number of boxed occurrences of 132 in a permutation. St001093The detour number of a graph. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the 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 St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001308The number of induced paths on three vertices in a graph. St001342The number of vertices in the center of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001374The Padmakar-Ivan index of a graph. St001391The disjunction number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001485The modular major index of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001593This is the number of standard Young tableaux of the given shifted shape. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001620The number of sublattices of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001657The number of twos in an integer partition. St001669The number of single rises in a Dyck path. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001695The natural comajor index of a standard Young tableau. St001696The natural major index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001754The number of tolerances of a finite lattice. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001792The arboricity of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the TurĆ”n graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001812The biclique partition number of a graph. St001885The number of binary words with the same proper border set. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001956The comajor index for set-valued two-row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000438The position of the last up step in a Dyck path. St000452The number of distinct eigenvalues of a graph. St000978The sum of the positions of double down-steps of a Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001619The number of non-isomorphic sublattices of a lattice. St001625The Mƶbius invariant of a lattice. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000977MacMahon's equal index of a Dyck path. St000981The length of the longest zigzag subpath. St000741The Colin de VerdiĆØre graph invariant. St000219The number of occurrences of the pattern 231 in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000338The number of pixed points of a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001586The number of odd parts smaller than the largest even part in an integer partition. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001651The Frankl number of a lattice. St000124The cardinality of the preimage of the Simion-Schmidt map. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001684The reduced word complexity of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000545The number of parabolic double cosets with minimal element being the given permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000509The diagonal index (content) of a partition. St001373The logarithm of the number of winning configurations of the lights out game on a graph. 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. St000467The hyper-Wiener index of a connected graph. St001720The minimal length of a chain of small intervals in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. 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 $eA$. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word. St001769The reflection length of a signed permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001964The interval resolution global dimension of a poset. St001563The value of the power-sum symmetric function evaluated at 1. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000264The girth of a graph, which is not a tree. 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 $K[x]/(x^n)$. 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. St000699The toughness times the least common multiple of 1,. St001423The number of distinct cubes in a binary word. St001570The minimal number of edges to add to make a graph Hamiltonian. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St001490The number of connected components of a skew partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Mƶbius function of a poset. St000137The Grundy value of an integer partition. St001527The cyclic permutation representation number of an integer partition. St000553The number of blocks of a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001768The number of reduced words of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St000902 The minimal number of repetitions of an integer composition. St001433The flag major index of a signed permutation. St001524The degree of symmetry of a binary word. St000691The number of changes of a binary word. St001817The number of flag weak exceedances of a signed permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Mƶbius function of a poset. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001395The number of strictly unfriendly partitions of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000449The number of pairs of vertices of a graph with distance 4. St000636The hull number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000917The open packing number of a graph. St000948The chromatic discriminant of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001367The smallest number which does not occur as degree of a vertex in a graph. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001654The monophonic hull number of a graph. St001672The restrained domination number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St000477The weight of a partition according to Alladi. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000510The number of invariant oriented cycles 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000225Difference between largest and smallest parts in a partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000674The number of hills of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000396The register function (or Horton-Strahler number) of a binary tree. St000679The pruning number of an ordered tree. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000385The number of vertices with out-degree 1 in a binary tree. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St001396Number of triples of incomparable elements in a finite poset. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000911The number of maximal antichains of maximal size in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. 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. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001571The Cartan determinant of the integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001608The number of coloured rooted trees 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001613The binary logarithm of the size of the center of a lattice. 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. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001763The Hurwitz number of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001881The number of factors of a lattice as a Cartesian product of lattices. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000846The maximal number of elements covering an element of a poset. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001621The number of atoms of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000474Dyson's crank of a partition. St000806The semiperimeter of the associated bargraph. St001857The number of edges in the reduced word graph of a signed permutation. St000657The smallest part of an integer composition. St000850The number of 1/2-balanced pairs in a poset. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000210Minimum over maximum difference of elements in cycles. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000253The crossing number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000761The number of ascents in an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000916The packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000990The first ascent of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001162The minimum jump of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 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 $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001256Number of simple reflexive modules that are 2-stable reflexive. St001272The number of graphs with the same degree sequence. St001274The number of indecomposable injective modules with projective dimension equal to two. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001344The neighbouring number of a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001638The book thickness of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000077The number of boxed and circled entries. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000102The charge of a semistandard tableau. St000105The number of blocks in the set partition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. 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. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000273The domination number of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000344The number of strongly connected outdegree sequences of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000401The size of the symmetry class of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000417The size of the automorphism group of the ordered tree. St000447The number of pairs of vertices of a graph with distance 3. St000469The distinguishing number of a graph. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000537The cutwidth of a graph. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000637The length of the longest cycle in a graph. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000778The metric dimension of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000823The number of unsplittable factors of the set partition. St000845The maximal number of elements covered by an element in a poset. St000903The number of different parts of an integer composition. St000943The number of spots the most unlucky car had to go further in a parking function. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001306The number of induced paths on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001322The size of a minimal independent dominating set in a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001381The fertility of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001705The number of occurrences of the pattern 2413 in a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001715The number of non-records in a permutation. St001716The 1-improper chromatic number of a graph. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001734The lettericity of a graph. St001736The total number of cycles in a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001829The common independence number of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000258The burning number of a graph. St000638The number of up-down runs of a permutation. St000652The maximal difference between successive positions of a permutation. St000680The Grundy value for Hackendot on posets. St001246The maximal difference between two consecutive entries of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001642The Prague dimension of a graph. St000226The convexity of a permutation. St000244The cardinality of the automorphism group of a graph. St000309The number of vertices with even degree. St000918The 2-limited packing number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St001409The maximal entry of a semistandard tableau.
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