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Your data matches 606 different statistics following compositions of up to 3 maps.
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Matching statistic: St001475
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 1
([],3)
=> 1
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> 1
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> 1
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 1
([],6)
=> 1
([(4,5)],6)
=> 1
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0).
Matching statistic: St000095
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001311
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St001336
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St001572
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The minimal number of edges to remove to make a graph bipartite.
Matching statistic: St001573
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([],1)
=> 0 = 1 - 1
([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 0 = 1 - 1
([],6)
=> 0 = 1 - 1
([(4,5)],6)
=> 0 = 1 - 1
Description
The minimal number of edges to remove to make a graph triangle-free.
The following 596 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000450The number of edges minus the number of vertices plus 2 of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000948The chromatic discriminant of a graph. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000535The rank-width of a graph. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001309The number of four-cliques in a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001638The book thickness of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000047The number of standard immaculate tableaux of a given shape. St000258The burning number of a graph. St000277The number of ribbon shaped standard tableaux. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000452The number of distinct eigenvalues of a graph. St000544The cop number of a graph. St000644The number of graphs with given frequency partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000778The metric dimension of a graph. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000918The 2-limited packing number of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. 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. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001642The Prague dimension of a graph. St001758The number of orbits of promotion on a graph. St001792The arboricity of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001949The rigidity index of a graph. St000089The absolute variation of a composition. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000469The distinguishing number of a graph. St000481The number of upper covers of a partition in dominance order. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000944The 3-degree of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001391The disjunction number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001673The degree of asymmetry of an integer composition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001777The number of weak descents in an integer composition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001871The number of triconnected components of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. 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. St000048The multinomial of the parts of a partition. St000079The number of alternating sign matrices for a given Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000537The cutwidth of a graph. St000549The number of odd partial sums of an integer partition. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. 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. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001007Number of simple modules with projective dimension 1 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. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001270The bandwidth of a graph. St001281The normalized isoperimetric number of a graph. St001282The number of graphs with the same chromatic polynomial. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001595The number of standard Young tableaux of the skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001644The dimension of a graph. St001675The number of parts equal to the part in the reversed composition. St001734The lettericity of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001746The coalition number of a graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. 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. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St000008The major index of the composition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000090The variation of a composition. St000091The descent variation of a composition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. 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. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. 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. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000369The dinv deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000636The hull number of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000676The number of odd rises of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000759The smallest missing part in an integer partition. St000761The number of ascents in an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001214The aft of an integer partition. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001252Half the sum of the even parts of a partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001485The modular major index of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001521Half the total irregularity of a graph. St001524The degree of symmetry of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001799The number of proper separations of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001883The mutual visibility number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000379The number of Hamiltonian cycles in a graph. St001964The interval resolution global dimension of a poset. St001108The 2-dynamic chromatic number of a graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001363The Euler characteristic of a graph according to Knill. 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. St001568The smallest positive integer that does not appear twice in the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001592The maximal number of simple paths between any two different vertices of a graph. St001651The Frankl number of a lattice. St000006The dinv of a Dyck path. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000668The least common multiple of the parts of the partition. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. 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. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000928The sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000455The second largest eigenvalue of a graph if it is integral. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St000656The number of cuts of a poset. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000340The number of non-final maximal constant sub-paths of length greater than one. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000567The sum of the products of all pairs of parts. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels 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. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000699The toughness times the least common multiple of 1,. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000744The length of the path to the largest entry in a standard Young tableau. St000792The Grundy value for the game of ruler on a binary word. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. 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. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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 $c_{n−1}$. St000984The number of boxes below precisely one peak. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g 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. 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. 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. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. 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. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000144The pyramid weight of the Dyck path. St000183The side length of the Durfee square of an integer partition. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000376The bounce deficit of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000969We 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 $c_{n-1}$. St000976The sum of the positions of double up-steps of a Dyck path. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001060The distinguishing index of a graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. 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 $eA$. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001413Half the length of the longest even length palindromic prefix of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001808The box weight or horizontal decoration of a Dyck path. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000997The even-odd crank of an integer partition. 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. St000264The girth of a graph, which is not a tree. St001890The maximum magnitude of the Möbius function of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001571The Cartan determinant of the integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000464The Schultz index of a connected graph. St000946The sum of the skew hook positions in a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001545The second Elser number of a connected graph. St000438The position of the last up step in a Dyck path. St001875The number of simple modules with projective dimension at most 1. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001763The Hurwitz number of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001095The number of non-isomorphic posets with precisely one further covering relation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of 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. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000782The indicator function of whether a given perfect matching is an L & P matching.
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