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Your data matches 558 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001333
(load all 14 compositions to match this statistic)

Mapping

St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

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: St001393
(load all 15 compositions to match this statistic)

Mapping

St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

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 14 compositions to match this statistic)

Mapping

St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1 = 0 + 1
([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => 2 = 1 + 1
([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([],4)
 => 1 = 0 + 1
([(2,3)],4)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1

Description

The Castelnuovo-Mumford regularity of a graph.

Matching statistic: St000010
(load all 4 compositions to match this statistic)

Mapping

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%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The length of the partition.

Matching statistic: St000159
(load all 10 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000160
(load all 4 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part

$spt(\lambda)$ of a partition

$\lambda$ . The sum

$spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences

$\begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*}$ analogous to those of the counting function of partitions, see [1] and [2].

Matching statistic: St000387
(load all 3 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000387: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2

Description

The matching number of a graph. For a graph

$G$ , this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in

$G$ .

Matching statistic: St000533
(load all 8 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

Matching statistic: St000548
(load all 3 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The number of different non-empty partial sums of an integer partition.

Matching statistic: St000783
(load all 8 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1

Description

The side length of the largest staircase partition fitting into a partition. For an integer partition

$(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer

$k$ such that

$\lambda_i > k-i$ for

$i\in\{1,\dots,k\}$ . In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.

The following 548 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001280The number of parts of an integer partition that are at least two. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001734The lettericity of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000340The number of non-final maximal constant sub-paths of length greater than one. St000378The diagonal inversion number of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000535The rank-width of a graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000897The number of different multiplicities of parts of an integer partition. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001271The competition number of a graph. St001512The minimum rank of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001871The number of triconnected components of a graph. St000287The number of connected components of a graph. St000544The cop number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001345The Hamming dimension of a graph. St001363The Euler characteristic of a graph according to Knill. St001471The magnitude of a Dyck path. St001642The Prague dimension of a graph. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St001814The number of partitions interlacing the given partition. St001108The 2-dynamic chromatic number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000011The number of touch points (or returns) of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000183The side length of the Durfee square of an integer partition. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000507The number of ascents of a standard tableau. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000628The balance of a binary word. St000665The number of rafts of a permutation. St000676The number of odd rises of a Dyck path. St000711The number of big exceedences of a permutation. St000778The metric dimension of a graph. St000834The number of right outer peaks of a permutation. St000920The logarithmic height of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. 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. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001657The number of twos in an integer partition. 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). St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. 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. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000381The largest part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000631The number of distinct palindromic decompositions of a binary word. St000636The hull number of a graph. St000644The number of graphs with given frequency partition. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000784The maximum of the length and the largest part of the integer partition. St000808The number of up steps of the associated bargraph. St000847The number of standard Young tableaux whose descent set is the binary word. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001389The number of partitions of the same length below the given integer partition. St001437The flex of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001955The number of natural descents for set-valued two row standard Young tableaux. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001391The disjunction number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St000741The Colin de Verdière graph invariant. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St000251The number of nonsingleton blocks of a set partition. St000442The maximal area to the right of an up step of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000260The radius of a connected graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000335The difference of lower and upper interactions. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000481The number of upper covers of a partition in dominance order. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000934The 2-degree of an integer partition. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension 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. St001128The exponens consonantiae of a partition. 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

$g$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001383The BG-rank of an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001672The restrained domination number of a graph. St001732The number of peaks visible from the left. 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. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001884The number of borders of a binary word. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000012The area of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000172The Grundy number of a graph. St000268The number of strongly connected orientations of a graph. St000293The number of inversions of a binary word. St000344The number of strongly connected outdegree sequences of a graph. St000382The first part of an integer composition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000452The number of distinct eigenvalues of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000658The number of rises of length 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St000905The number of different multiplicities of parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic 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. St001733The number of weak left to right maxima of a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001963The tree-depth of a graph. St000259The diameter of a connected graph. St000331The number of upper interactions of a 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. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001272The number of graphs with the same degree sequence. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001282The number of graphs with the same chromatic polynomial. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001518The number of graphs with the same ordinary spectrum as the given graph. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000454The largest eigenvalue of a graph if it is integral. 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

$c_{n−1}$ . St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000455The second largest eigenvalue of a graph if it is integral. St000762The sum of the positions of the weak records of an integer composition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset. St000553The number of blocks of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001568The smallest positive integer that does not appear twice in the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001890The maximum magnitude of the Möbius function of a poset. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000735The last entry on the main diagonal of a standard tableau. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000420The number of Dyck paths that are weakly above a Dyck path. St000874The position of the last double rise in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001520The number of strict 3-descents. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001808The box weight or horizontal decoration of a Dyck path. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001645The pebbling number of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000456The monochromatic index of a connected graph. St000914The sum of the values of the Möbius function of a poset. St001592The maximal number of simple paths between any two different vertices of a graph. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000088The row sums of the character table of the symmetric group. St000148The number of odd parts of a partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000475The number of parts equal to 1 in a partition. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000549The number of odd partial sums of an integer partition. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. 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. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. 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. St000983The length of the longest alternating subword. 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001118The acyclic chromatic index of a graph. St001129The product of the squares of the parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001313The number of Dyck paths above the lattice path given by a binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. 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. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001484The number of singletons of an integer partition. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001838The number of nonempty primitive factors of a binary word. St001877Number of indecomposable injective modules with projective dimension 2. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000928The sum of the coefficients of the character polynomial of an integer partition. St001651The Frankl number of a lattice. St000460The hook length of the last cell along the main diagonal of an integer partition. St000699The toughness times the least common multiple of 1,. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000422The energy of a graph, if it is integral. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. 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. St001845The number of join irreducibles minus the rank of a lattice. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000744The length of the path to the largest entry in a standard Young tableau. St000939The number of characters of the symmetric group whose value on the partition is positive. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001626The number of maximal proper sublattices of a lattice. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given 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.