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Your data matches 820 different statistics following compositions of up to 3 maps.
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Matching statistic: St001333
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The cardinality of a minimal edge-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 only the graph with one edge.
Matching statistic: St001340
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The cardinality of a minimal non-edge 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 only the graph with two isolated vertices.
Matching statistic: St001393
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St001261
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3)],6)
=> 4 = 1 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> 3 = 0 + 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 3 = 0 + 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> 3 = 0 + 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 3 = 0 + 3
([(1,6),(2,5),(3,4)],7)
=> 4 = 1 + 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> 4 = 1 + 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> 3 = 0 + 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 3 = 0 + 3
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 3 = 0 + 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 3 = 0 + 3
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> 3 = 0 + 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 4 = 1 + 3
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> 3 = 0 + 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 3 = 0 + 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> 3 = 0 + 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001568
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001568: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> 1 = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> 1 = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 2 = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> 2 = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> 1 = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> 1 = 0 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> 1 = 0 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> 1 = 0 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> 2 = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> 1 = 0 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> 1 = 0 + 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 1 = 0 + 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> 1 = 0 + 1
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,2]
=> 1 = 0 + 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,3]
=> 1 = 0 + 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3]
=> 1 = 0 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,2]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,3]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3]
=> 2 = 0 + 2
Description
The length of the partition.
Matching statistic: St000473
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St001280
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> 2 = 0 + 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001354
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3 = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 2 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> 2 = 0 + 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The number of series nodes in the modular decomposition of a graph.
Matching statistic: St000143
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [3]
=> 0
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> [3]
=> 0
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,2]
=> [2]
=> 0
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,3]
=> [3]
=> 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3]
=> [3]
=> 0
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
The following 810 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000929The constant term of the character polynomial of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000048The multinomial of the parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000346The number of coarsenings of a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. 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. 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. 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001251The number of parts of a partition that are not congruent 1 modulo 3. 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. St001732The number of peaks visible from the left. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St000024The number of double up and double down steps of a Dyck path. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000668The least common multiple of the parts of the partition. St000733The row containing the largest entry of a standard tableau. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001512The minimum rank of a graph. St000013The height of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000708The product of the parts of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001352The number of internal nodes in the modular decomposition of a graph. St000017The number of inversions of a standard tableau. St000057The Shynar inversion number of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000478Another weight of a partition according to Alladi. St000480The number of lower covers of a partition in dominance order. 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. 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. St000649The number of 3-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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. St000761The number of ascents in an integer composition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001010Number of indecomposable injective modules with projective dimension g-1 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001423The number of distinct cubes in a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001484The number of singletons of an integer partition. St001513The number of nested exceedences of a permutation. St001541The Gini index of an integer partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001557The number of inversions of the second entry of a permutation. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001596The number of two-by-two squares inside a skew partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001675The number of parts equal to the part in the reversed composition. St001728The number of invisible descents of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000025The number of initial rises of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000184The size of the centralizer of any permutation of given cycle type. 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. St000223The number of nestings in the permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000392The length of the longest run of ones in a binary word. St000409The number of pitchforks in a binary tree. St000617The number of global maxima of a Dyck path. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000703The number of deficiencies of a permutation. St000707The product of the factorials of the parts. St000764The number of strong records in an integer composition. 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000913The number of ways to refine the partition into singletons. St000920The logarithmic height of a Dyck path. St000921The number of internal inversions of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension 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. 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. St001038The minimal height of a column in the parallelogram polyomino 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. 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. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001184Number of indecomposable injective modules with grade at least 1 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. 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. 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. St001424The number of distinct squares in a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. 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. St001597The Frobenius rank of a skew partition. St001665The number of pure excedances of a permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001777The number of weak descents in an integer composition. St001801Half the number of preimage-image pairs of different parity in a permutation. 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). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000293The number of inversions of a binary word. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000439The position of the first down step of a Dyck path. St000507The number of ascents of a standard tableau. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000820The number of compositions obtained by rotating the composition. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000982The length of the longest constant subword. St000996The number of exclusive left-to-right maxima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. 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:
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 $eA$. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001439The number of even weak deficiencies and of odd weak exceedences. St001471The magnitude of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001814The number of partitions interlacing the given partition. St000009The charge of a standard tableau. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000539The number of odd inversions of a permutation. St000702The number of weak deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000991The number of right-to-left minima of a permutation. 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. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001589The nesting number of a perfect matching. 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. St000094The depth of an ordered tree. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000495The number of inversions of distance at most 2 of a permutation. St000521The number of distinct subtrees of an ordered tree. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of 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. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000494The number of inversions of distance at most 3 of a permutation. St000258The burning number of a graph. St001358The largest degree of a regular subgraph of a graph. St001462The number of factors of a standard tableaux under concatenation. St000383The last part of an integer composition. St000916The packing number of a graph. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000382The first part of an integer composition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in 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. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000315The number of isolated vertices of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001691The number of kings in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000287The number of connected components of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000117The number of centered tunnels of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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 $g$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000273The domination number of a graph. St000544The cop number of a graph. St001829The common independence number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000286The number of connected components of the complement of a graph. St000553The number of blocks of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000075The orbit size of a standard tableau under promotion. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001644The dimension of a graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000335The difference of lower and upper interactions. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000989The number of final rises of a permutation. St000461The rix statistic of a permutation. St001826The maximal number of leaves on a vertex of a graph. St000387The matching number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001307The number of induced stars on four vertices in a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000918The 2-limited packing number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000309The number of vertices with even degree. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057The Grundy value of the game of creating an independent set in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001374The Padmakar-Ivan index of a graph. St001386The number of prime labellings of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity 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. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. 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. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001871The number of triconnected components of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000455The second largest eigenvalue of a graph if it is integral. St000535The rank-width of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. 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. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St001734The lettericity of a graph. St001743The discrepancy of a graph. St001765The number of connected components of the friends and strangers graph. 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. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001093The detour number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001703The villainy of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000699The toughness times the least common multiple of 1,. St001060The distinguishing index of a graph. St001637The number of (upper) dissectors of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000758The length of the longest staircase fitting into an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000386The number of factors DDU in a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000301The number of facets of the stable set polytope of a graph. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001371The length of the longest Yamanouchi prefix of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000627The exponent of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001877Number of indecomposable injective modules with projective dimension 2. St001884The number of borders of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. 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. St001875The number of simple modules with projective dimension at most 1. St000741The Colin de Verdière graph invariant. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000366The number of double descents of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000475The number of parts equal to 1 in a partition. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000546The number of global descents of a permutation. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000031The number of cycles in the cycle decomposition 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. St000007The number of saliances of the permutation. St000862The number of parts of the shifted shape of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001383The BG-rank of an integer partition. St001395The number of strictly unfriendly partitions of a graph. St001561The value of the elementary symmetric function evaluated at 1. St001638The book thickness of a graph. St001828The Euler characteristic of a graph. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000482The (zero)-forcing number of a graph. St000537The cutwidth of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St001270The bandwidth of a graph. St001281The normalized isoperimetric number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001625The Möbius invariant of a lattice. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000081The number of edges of a graph. St000146The Andrews-Garvan crank of a partition. St000171The degree of the graph. St000271The chromatic index of a graph. St000312The number of leaves in a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001112The 3-weak dynamic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001286The annihilation number of a graph. St001345The Hamming dimension of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001391The disjunction number of a graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001716The 1-improper chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001869The maximum cut size of a graph. St001883The mutual visibility number of a graph. St000086The number of subgraphs. St000454The largest eigenvalue of a graph if it is integral. St000468The Hosoya index of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St001110The 3-dynamic chromatic number of a graph. St001277The degeneracy of a graph. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001645The pebbling number of a connected graph. St001754The number of tolerances of a finite lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000479The Ramsey number of a graph. St000090The variation of a composition. St000091The descent variation of a composition. St000252The number of nodes of degree 3 of a binary tree. St000302The determinant of the distance matrix of a connected graph. St000344The number of strongly connected outdegree sequences of a graph. St000379The number of Hamiltonian cycles in a graph. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001470The cyclic holeyness of a permutation. St001736The total number of cycles in a graph. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000023The number of inner peaks of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000272The treewidth of a graph. St000338The number of pixed points of a permutation. St000353The number of inner valleys of a permutation. St000362The size of a minimal vertex cover of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000657The smallest part of an integer composition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000779The tier of a permutation. St000805The number of peaks of the associated bargraph. St000831The number of indices that are either descents or recoils. St000872The number of very big descents of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000948The chromatic discriminant of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001501The dominant dimension of magnitude 1 Nakayama algebras. St001592The maximal number of simple paths between any two different vertices of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001928The number of non-overlapping descents in a permutation. St000087The number of induced subgraphs. St000092The number of outer peaks of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000456The monochromatic index of a connected graph. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000873The aix statistic of a permutation. St000926The clique-coclique number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001963The tree-depth of a graph. St000209Maximum difference of elements in cycles. St000300The number of independent sets of vertices of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000653The last descent of a permutation. St000922The minimal number such that all substrings of this length are unique. St000956The maximal displacement of a permutation. St000981The length of the longest zigzag subpath. St001235The global dimension of the corresponding Comp-Nakayama 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. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001778The largest greatest common divisor of an element and its image in a permutation. St000144The pyramid weight of the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001706The number of closed sets in a graph. St000030The sum of the descent differences of a permutations. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001180Number of indecomposable injective modules with projective dimension at most 1. St000156The Denert index of a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001726The number of visible inversions of a permutation. St000393The number of strictly increasing runs in a binary word. St000795The mad of a permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001834The number of non-isomorphic minors of a graph. St001545The second Elser number of a connected graph. St000467The hyper-Wiener index of a connected graph. St000464The Schultz index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001260The permanent of an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St000311The number of vertices of odd degree in a graph. St000890The number of nonzero entries in an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001649The length of a longest trail in a graph. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001757The number of orbits of toric promotion on a graph. St001783The number of odd automorphisms of a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St000343The number of spanning subgraphs of a graph.
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