Your data matches 235 different statistics following compositions of up to 3 maps.
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Matching statistic: St000076
Mapping
St000076: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 0 = 1 - 1
[[1,0],[0,1]]
 => 0 = 1 - 1
[[0,1],[1,0]]
 => 1 = 2 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => 0 = 1 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => 1 = 2 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 2 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => 2 = 3 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 4 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => 4 = 5 - 1
Description
The rank of the alternating sign matrix in the alternating sign matrix poset. This rank is the sum of the entries of the monotone triangle minus $\binom{n+2}{3}$, which is the smallest sum of the entries in the set of all monotone triangles with bottom row $1\dots n$. Alternatively, $rank(A)=\frac{1}{2} \sum_{i,j=1}^n (i-j)^2 a_{ij}$, see [3, thm.5.1].
Matching statistic: St000020
Mapping
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000020: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1,0]
 => [1,0]
 => [1] => 1
[[1,0],[0,1]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 1
[[0,1],[1,0]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [2,1] => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => 5
[[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => 4
Description
The rank of the permutation. This is its position among all permutations of the same size ordered lexicographically. This can be computed using the Lehmer code of a permutation: $$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$ where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.
Matching statistic: St000093
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => ([],1)
 => ([],1)
 => 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => 5
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => ([],1)
 => [1]
 => 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => [1]
 => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
Description
The largest part of an integer partition.
Matching statistic: St000305
Mapping
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1,0]
 => [1,0]
 => [2,1] => 1
[[1,0],[0,1]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 1
[[0,1],[1,0]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 5
[[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 4
Description
The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000384
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => ([],1)
 => [1]
 => 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => [1]
 => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
Description
The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].
Matching statistic: St000784
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => ([],1)
 => [1]
 => 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => [1]
 => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St000798
Mapping
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1,0]
 => [1,0]
 => [2,1] => 1
[[1,0],[0,1]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 1
[[0,1],[1,0]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 5
[[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
Description
The makl of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(1\underline{32})$, $(\underline{31}2)$, $(\underline{32}1)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000833
Mapping
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1,0]
 => [1,0]
 => [2,1] => 1
[[1,0],[0,1]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 1
[[0,1],[1,0]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 2
[[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 5
[[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
Description
The comajor index of a permutation. This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St000355
Mapping
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[[1,0],[0,1]]
 => [[1,1],[2]]
 => [3,1,2] => [3,1,2] => 0 = 1 - 1
[[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [6,4,5,1,2,3] => [5,1,2,6,4,3] => 1 = 2 - 1
[[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 3 = 4 - 1
[[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [5,4,6,1,2,3] => [6,1,2,5,4,3] => 0 = 1 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => [5,3,6,1,2,4] => [6,1,5,3,2,4] => 1 = 2 - 1
[[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [4,3,5,1,2,6] => [5,1,4,3,2,6] => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [5,2,6,1,3,4] => [6,5,2,1,3,4] => 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [4,2,5,1,3,6] => [5,4,2,1,3,6] => 4 = 5 - 1
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
The following 225 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000359The number of occurrences of the pattern 23-1. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001956The comajor index for set-valued two-row standard Young tableaux. St000189The number of elements in the poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000087The number of induced subgraphs. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number 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. St001093The detour number of a graph. St001286The annihilation 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. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001616The number of neutral elements in a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in 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. St001720The minimal length of a chain of small intervals in a lattice. St001746The coalition number of a graph. St001820The size of the image of the pop stack sorting operator. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000081The number of edges of a graph. St000259The diameter of a connected graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001391The disjunction number of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001619The number of non-isomorphic sublattices of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001649The length of a longest trail in a graph. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001827The number of two-component spanning forests of a graph. St001834The number of non-isomorphic minors of a graph. St001846The number of elements which do not have a complement in the lattice. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001378The product of the cohook lengths of the integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001885The number of binary words with the same proper border set. St000770The major index of an integer partition when read from bottom to top. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001271The competition number of a graph. St001330The hat guessing number of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000327The number of cover relations in a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001118The acyclic chromatic index of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000260The radius of a connected graph. St000881The number of short braid edges in the graph of braid moves of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000958The number of Bruhat factorizations of a permutation. St001060The distinguishing index of a graph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001948The number of augmented double ascents of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000115The single entry in the last row. St000181The number of connected components of the Hasse diagram for the poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000456The monochromatic index of a connected graph. St000545The number of parabolic double cosets with minimal element being the given permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000942The number of critical left to right maxima of the parking functions. St000981The length of the longest zigzag subpath. St000993The multiplicity of the largest part of an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001500The global dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001531Number of partial orders contained in the poset determined by the Dyck path. St001545The second Elser number of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001722The number of minimal chains with small intervals between a binary word and the top element. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001808The box weight or horizontal decoration of a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001863The number of weak excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001959The product of the heights of the peaks of a Dyck path. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000879The number of long braid edges in the graph of braid moves of a permutation. 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 $K[x]/(x^n)$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.