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Your data matches 953 different statistics following compositions of up to 3 maps.
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Matching statistic: St001022
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001137
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001167
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001311
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001319
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001319: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001319: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of isolated vertices and a star if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[6] => ([],6)
=> 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[1,6] => ([(5,6)],7)
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[7] => ([],7)
=> 0
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
The following 943 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001336The minimal number of vertices in a graph whose complement is triangle-free. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000052The number of valleys of a Dyck path not on the x-axis. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000731The number of double exceedences of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000877The depth of the binary word interpreted as a path. St000944The 3-degree of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001172The number of 1-rises at odd height of a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001512The minimum rank of a graph. 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. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000048The multinomial of the parts of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000346The number of coarsenings of a partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. 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. St000920The logarithmic height of a Dyck path. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001261The Castelnuovo-Mumford regularity of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001917The order of toric promotion on the set of labellings of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000024The number of double up and double down steps of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000065The number of entries equal to -1 in an alternating sign matrix. St000091The descent variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000169The cocharge of a standard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000313The number of degree 2 vertices of a graph. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing 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 $3$. St000386The number of factors DDU in a Dyck path. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000496The rcs statistic of a set partition. St000534The number of 2-rises of a permutation. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000648The number of 2-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001160The number of proper blocks (or intervals) of a permutations. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001214The aft of an integer partition. St001252Half the sum of the even parts of a partition. St001271The competition number of a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001469The holeyness of a permutation. St001485The modular major index of a binary word. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001911A descent variant minus the number of inversions. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000092The number of outer peaks of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000201The number of leaf nodes in a binary tree. St000255The number of reduced Kogan faces with the permutation as type. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000382The first part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000413The number of ordered trees with the same underlying unordered tree. St000522The number of 1-protected nodes of a rooted tree. St000630The length of the shortest palindromic decomposition of a binary word. St000669The number of permutations obtained by switching ascents or descents of size 2. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000733The row containing the largest entry of a standard tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. 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. St000816The number of standard composition tableaux of the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001151The number of blocks with odd minimum. St001313The number of Dyck paths above the lattice path given by a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001732The number of peaks visible from the left. St001741The largest integer such that all patterns of this size are contained in the permutation. 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. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000676The number of odd rises of a Dyck path. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000353The number of inner valleys of a permutation. St000369The dinv deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 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. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000538The number of even inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. 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. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000836The number of descents of distance 2 of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001596The number of two-by-two squares inside a skew partition. St000078The number of alternating sign matrices whose left key is the permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000886The number of permutations with the same antidiagonal sums. St001597The Frobenius rank of a skew partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000299The number of nonisomorphic vertex-induced subtrees. St000002The number of occurrences of the pattern 123 in a permutation. 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. St000883The number of longest increasing subsequences of a permutation. St001642The Prague dimension of a graph. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000039The number of crossings of a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000663The number of right floats of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000871The number of very big ascents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001735The number of permutations with the same set of runs. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001731The factorization defect of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000352The Elizalde-Pak rank of a permutation. St000619The number of cyclic descents of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000312The number of leaves in a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000079The number of alternating sign matrices for a given Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000948The chromatic discriminant of a graph. St000959The number of strong Bruhat factorizations of a permutation. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000089The absolute variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000650The number of 3-rises of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000769The major index of a composition regarded as a word. St000864The number of circled entries of the shifted recording tableau of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001307The number of induced stars on four vertices in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001673The degree of asymmetry of an integer composition. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001871The number of triconnected components of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000047The number of standard immaculate tableaux of a given shape. St000099The number of valleys of a permutation, including the boundary. St000277The number of ribbon shaped standard tableaux. St000340The number of non-final maximal constant sub-paths of length greater than one. St000381The largest part of an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000537The cutwidth of a graph. St000767The number of runs in an integer composition. St000778The metric dimension of a graph. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000903The number of different parts of an integer composition. St000988The orbit size of a permutation under Foata's bijection. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001270The bandwidth of a graph. St001286The annihilation number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001644The dimension of a graph. St001758The number of orbits of promotion on a graph. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001962The proper pathwidth of a graph. St000636The hull number of a graph. St001471The magnitude of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000837The number of ascents of distance 2 of a permutation. St000961The shifted major index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001220The width of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000806The semiperimeter of the associated bargraph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001592The maximal number of simple paths between any two different vertices of a graph. St001933The largest multiplicity of a part in an integer partition. St000910The number of maximal chains of minimal length in a poset. St000656The number of cuts of a poset. St000010The length of the partition. St000012The area of a Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000378The diagonal inversion number of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. 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. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000992The alternating sum of the parts of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001127The sum of the squares of the parts of a partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001388The number of non-attacking neighbors of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001424The number of distinct squares in a binary word. St001484The number of singletons of an integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001541The Gini index of an integer partition. St001549The number of restricted non-inversions between exceedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001910The height of the middle non-run of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St000026The position of the first return 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. St000053The number of valleys of the Dyck path. St000060The greater neighbor of the maximum. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000288The number of ones in a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000327The number of cover relations in a poset. St000335The difference of lower and upper interactions. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000402Half the size of the symmetry class of a permutation. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000443The number of long tunnels of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000530The number of permutations with the same descent word as the given permutation. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000939The number of characters of the symmetric group whose value on the partition is positive. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001432The order dimension of the partition. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001637The number of (upper) dissectors of a poset. St001780The order of promotion on the set of standard tableaux of given shape. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000294The number of distinct factors of a binary word. St000439The position of the first down step of a Dyck path. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000933The number of multipartitions of sizes given by an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001074The number of inversions of the cyclic embedding of a permutation. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
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. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000219The number of occurrences of the pattern 231 in a permutation. St001128The exponens consonantiae of a partition. St000045The number of linear extensions of a binary tree. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000789The number of crossing-similar perfect matchings of a perfect matching. St001282The number of graphs with the same chromatic polynomial. St000649The number of 3-excedences of a permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000958The number of Bruhat factorizations of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000881The number of short braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001513The number of nested exceedences of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001964The interval resolution global dimension of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000880The number of connected components of long braid edges in the graph of braid moves 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. St001114The number of odd descents of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000542The number of left-to-right-minima of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001281The normalized isoperimetric number of a graph. St000145The Dyson rank of a partition. 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. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000699The toughness times the least common multiple of 1,. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001557The number of inversions of the second entry of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001811The Castelnuovo-Mumford regularity of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000478Another weight of a partition according to Alladi. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs 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. St001808The box weight or horizontal decoration of a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000015The number of peaks of a Dyck path. St000438The position of the last up step in a Dyck path. St000477The weight of a partition according to Alladi. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000937The number of positive values of the symmetric group character corresponding to the partition. St001060The distinguishing index of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001530The depth of a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001866The nesting alignments of a signed permutation. 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 $eA$. St000260The radius of a connected graph. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001520The number of strict 3-descents. St001822The number of alignments of a signed permutation. St001948The number of augmented double ascents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000456The monochromatic index of a connected graph. St000707The product of the factorials of the parts. St001531Number of partial orders contained in the poset determined by the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001959The product of the heights of the peaks of a Dyck path. St000464The Schultz index of a connected graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001545The second Elser number of a connected graph. St000741The Colin de Verdière graph invariant. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001875The number of simple modules with projective dimension at most 1. 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. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001763The Hurwitz number of an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000146The Andrews-Garvan crank of a partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000629The defect of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000995The largest even part of an integer partition. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001695The natural comajor 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. St001712The number of natural descents of a standard Young tableau. 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. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000088The row sums of the character table of the symmetric group. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000331The number of upper interactions of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000644The number of graphs with given frequency partition. St000675The number of centered multitunnels of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000792The Grundy value for the game of ruler on a binary word. 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. 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. St000913The number of ways to refine the partition into singletons. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001595The number of standard Young tableaux of the skew partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001838The number of nonempty primitive factors of a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001885The number of binary words with the same proper border set. St000264The girth of a graph, which is not a tree. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001470The cyclic holeyness of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000100The number of linear extensions of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. 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. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001344The neighbouring number of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St000137The Grundy value of an integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St000474Dyson's crank of a partition. St000936The number of even values of the symmetric group character corresponding to 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. St000284The Plancherel distribution on integer partitions. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps 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. St000674The number of hills of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000981The length of the longest zigzag subpath. St000102The charge of a semistandard tableau. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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