Your data matches 403 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001595
(load all 4 compositions to match this statistic)
Mapping
St001595: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => 1 = 0 + 1
[[1,1],[]]
 => 1 = 0 + 1
[[2,1],[1]]
 => 2 = 1 + 1
[[3],[]]
 => 1 = 0 + 1
[[2,1],[]]
 => 2 = 1 + 1
[[3,1],[1]]
 => 3 = 2 + 1
[[2,2],[1]]
 => 2 = 1 + 1
[[3,2],[2]]
 => 3 = 2 + 1
[[1,1,1],[]]
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => 3 = 2 + 1
[[2,1,1],[1]]
 => 3 = 2 + 1
[[4],[]]
 => 1 = 0 + 1
[[3,1],[]]
 => 3 = 2 + 1
[[2,2],[]]
 => 2 = 1 + 1
[[2,1,1],[]]
 => 3 = 2 + 1
[[3,3],[2]]
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => 1 = 0 + 1
[[5],[]]
 => 1 = 0 + 1
[[1,1,1,1,1],[]]
 => 1 = 0 + 1
[[6],[]]
 => 1 = 0 + 1
[[1,1,1,1,1,1],[]]
 => 1 = 0 + 1
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St000849
(load all 5 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The number of 1/3-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
Number of pairs of incomparable elements in a finite poset. For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000100
(load all 3 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => 2 = 1 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2 = 1 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The number of linear extensions of a poset.
Matching statistic: St001268
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => 2 = 1 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2 = 1 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The size of the largest ordinal summand in the poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
Matching statistic: St001779
(load all 2 compositions to match this statistic)
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
St001779: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[2,1],[1]]
 => ([],2)
 => 2 = 1 + 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2 = 1 + 1
[[3,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[3,2],[2]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[2,1,1],[1]]
 => ([(1,2)],3)
 => 3 = 2 + 1
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3 = 2 + 1
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
Description
The order of promotion on the set of linear extensions of a poset.
Matching statistic: St000081
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([(0,1)],2)
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The number of edges of a graph.
Matching statistic: St000145
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => [1]
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => [1]
 => 0
[[2,1],[1]]
 => ([],2)
 => [2]
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => [2]
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => [3]
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => [2]
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => [3]
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => [3]
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => [3]
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [1]
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [1]
 => 0
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000171
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([(0,1)],2)
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000271
Mapping
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[2],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[2,1],[1]]
 => ([],2)
 => ([(0,1)],2)
 => 1
[[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
[[3,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[3,2],[2]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[2,2,1],[1,1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[2,1,1],[1]]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
[[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0
Description
The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.
The following 393 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001649The length of a longest trail in a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000086The number of subgraphs. St000088The row sums of the character table of the symmetric group. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000384The maximal part of the shifted composition of an integer partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000784The maximum of the length and the largest part of the integer partition. 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. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000935The number of ordered refinements of an integer partition. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001725The harmonious chromatic number of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000377The dinv defect of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001176The size of a partition minus its first part. 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000010The length of the partition. St000071The number of maximal chains in a poset. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000346The number of coarsenings of a partition. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000867The sum of the hook lengths in the first row of an integer partition. St000909The number of maximal chains of maximal size in a poset. St000933The number of multipartitions of sizes given by an integer partition. St001127The sum of the squares of the parts of a partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000095The number of triangles of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000454The largest eigenvalue of a graph if it is integral. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001118The acyclic chromatic index of a graph. St000477The weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000075The orbit size of a standard tableau under promotion. St000259The diameter of a connected graph. St000264The girth of a graph, which is not a tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000294The number of distinct factors of a binary word. St000411The tree factorial of a binary tree. St000412The number of binary trees with the same underlying unordered tree. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. 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. St000842The breadth of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a 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 $eA$. 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 $eA$. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001516The number of cyclic bonds of a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001885The number of binary words with the same proper border set. St000045The number of linear extensions of a binary tree. St000060The greater neighbor of the maximum. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000296The length of the symmetric border of a binary word. St000327The number of cover relations in a poset. 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. St000520The number of patterns in a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. St000568The hook number of a binary tree. St000569The sum of the heights of the vertices of a binary tree. St000570The Edelman-Greene number of a permutation. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000633The size of the automorphism group of a poset. St000634The number of endomorphisms of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000639The number of relations in a poset. St000640The rank of the largest boolean interval in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000690The size of the conjugacy class of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000806The semiperimeter of the associated bargraph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000914The sum of the values of the Möbius function of a poset. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. 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. St001437The flex of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. 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. St001884The number of borders of a binary word. St001890The maximum magnitude of the Möbius function of a poset. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000219The number of occurrences of the pattern 231 in a permutation. St000226The convexity of a permutation. 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. St000295The length of the border of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. 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. St000462The major index minus the number of excedences of 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. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St000519The largest length of a factor maximising the subword complexity. St000538The number of even inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. 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. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. 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. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. 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. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. 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. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000629The defect of a binary word. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001388The number of non-attacking neighbors of 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. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001930The weak major index of a binary word. St001960The number of descents of a permutation minus one if its first entry is not one. 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 $K[x]/(x^n)$. 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. St000934The 2-degree of an integer partition. St000707The product of the factorials of the parts. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition.