- FindStat

Your data matches 531 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000267
(load all 18 compositions to match this statistic)

Mapping

St000267: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of maximal spanning forests contained in a graph. A maximal spanning forest in a graph is a maximal acyclic subgraph. In other words, a spanning forest is a union of spanning trees in all connected components. See also [1] for this and further definitions. For connected graphs, this is the same as [[St000096]].

Matching statistic: St001546
(load all 18 compositions to match this statistic)

Mapping

St001546: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of monomials in the Tutte polynomial of a graph.

Matching statistic: St001351
(load all 7 compositions to match this statistic)

Mapping

St001351: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Albertson index of a graph. This is

$\sum_{\{u,v\}\in E} |d(u)-d(v)|$ , where

$E$ is the set of edges and

$d_v$ is the degree of vertex

$v$ , see [1]. In particular, this statistic vanishes on graphs whose components are all regular, see [2].

Matching statistic: St001374
(load all 7 compositions to match this statistic)

Mapping

St001374: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Padmakar-Ivan index of a graph. For an edge

$e=(u, v)$ , let

$n_{e, u}$ be the number of edges in a graph

$G$ induced by the set of vertices

$\{w: d(u, w) < d(v, w)\}$ , where

$d(u,v)$ denotes the distance between

$u$ and

$v$ . Then the PI-index of

$G$ is

$\sum_{e=(u,v)} n_{e, u} + n_{e, v}.$

Matching statistic: St001690
(load all 18 compositions to match this statistic)

Mapping

St001690: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. Put differently, for every vertex

$v$ of such a path

$P$ , there is a vertex

$w\in P$ and a vertex

$u\not\in P$ such that

$(v, u)$ and

$(u, w)$ are edges. The length of such a path is

$0$ if the graph is a forest. It is maximal, if and only if the graph is obtained from a graph

$H$ with a Hamiltonian path by joining a new vertex to each of the vertices of

$H$ .

Matching statistic: St000049
(load all 9 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000049: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of set partitions whose sorted block sizes correspond to the partition.

Matching statistic: St000086

Mapping

Mp00274: Graphs —block-cut tree⟶ Graphs
St000086: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of subgraphs. Given a graph

$G$ , this is the number of graphs

$H$ such that

$H \hookrightarrow G$ .

Matching statistic: St000096
(load all 4 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000096: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of spanning trees of a graph. A subgraph

$H \subseteq G$ is a spanning tree if

$V(H)=V(G)$ and

$H$ is a tree (i.e.

$H$ is connected and contains no cycles).

Matching statistic: St000097
(load all 4 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098
(load all 4 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

The following 521 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000172The Grundy number of a graph. St000271The chromatic index of a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000349The number of different adjacency matrices of a graph. St000363The number of minimal vertex covers of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000517The Kreweras number of an integer partition. St000722The number of different neighbourhoods in a graph. St000763The sum of the positions of the strong records of an integer composition. 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. St000822The Hadwiger number of the graph. St000935The number of ordered refinements of an integer partition. St001029The size of the core of a graph. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001386The number of prime labellings of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001694The number of maximal dissociation sets in a graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001725The harmonious chromatic number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001883The mutual visibility number of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St000081The number of edges of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000362The size of a minimal vertex cover of a graph. St000403The Szeged index minus the Wiener index of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000465The first Zagreb index of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000571The F-index (or forgotten topological index) of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000846The maximal number of elements covering an element of a poset. St000915The Ore degree of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000995The largest even part of an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001522The total irregularity of a graph. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001743The discrepancy of a graph. St001764The number of non-convex subsets of vertices in a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001962The proper pathwidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000983The length of the longest alternating subword. St001500The global dimension of magnitude 1 Nakayama algebras. St001620The number of sublattices of a lattice. St001706The number of closed sets in a graph. St001757The number of orbits of toric promotion on a graph. St001762The number of convex subsets of vertices in a graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St000003The number of standard Young tableaux of the partition. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000087The number of induced subgraphs. St000088The row sums of the character table of the symmetric group. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000469The distinguishing number of a graph. St000525The number of posets with the same zeta polynomial. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000926The clique-coclique number of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. 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. St001312Number of parabolic noncrossing partitions indexed by the composition. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. 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. St001395The number of strictly unfriendly partitions of a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001675The number of parts equal to the part in the reversed composition. 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. St001746The coalition number of a graph. St001763The Hurwitz number of an integer partition. 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. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001813The product of the sizes of the principal order filters in a poset. St001827The number of two-component spanning forests of a graph. St001829The common independence number of a graph. St001833The number of linear intervals in a lattice. St001838The number of nonempty primitive factors of a binary word. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001901The largest multiplicity of an irreducible representation 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. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000008The major index of the composition. St000095The number of triangles of a graph. St000225Difference between largest and smallest parts in a partition. St000268The number of strongly connected orientations of a graph. St000295The length of the border of a binary word. St000302The determinant of the distance matrix of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000327The number of cover relations in a poset. St000344The number of strongly connected outdegree sequences of a graph. St000377The dinv defect of an integer partition. St000448The number of pairs of vertices of a graph with distance 2. St000467The hyper-Wiener index of a connected graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000845The maximal number of elements covered by an element in a poset. St000979Half of MacMahon's equal index of a Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001350Half of the Albertson index of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001521Half the total irregularity of a graph. St001541The Gini index of an integer partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001689The number of celebrities in a graph. St001696The natural major index of a standard Young tableau. St001742The difference of the maximal and the minimal degree in a graph. St001777The number of weak descents in an integer composition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000418The number of Dyck paths that are weakly below a Dyck path. St000644The number of graphs with given frequency partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of a permutation. St000922The minimal number such that all substrings of this length are unique. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000982The length of the longest constant subword. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001959The product of the heights of the peaks of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000055The inversion sum of a permutation. St000154The sum of the descent bottoms of a permutation. St000160The multiplicity of the smallest part of a partition. St000163The size of the orbit of the set partition under rotation. St000217The number of occurrences of the pattern 312 in a permutation. St000258The burning number of a graph. St000315The number of isolated vertices of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000416The number of inequivalent increasing trees of an ordered tree. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000490The intertwining number of a set partition. St000519The largest length of a factor maximising the subword complexity. St000543The size of the conjugacy class of a binary word. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000657The smallest part of an integer composition. 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. St000692Babson and Steingrímsson's statistic of a permutation. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000738The first entry in the last row of a standard tableau. St000762The sum of the positions of the weak records of an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000795The mad of a permutation. St000808The number of up steps of the associated bargraph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in 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,. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001564The value of the forgotten symmetric functions when all variables set to 1. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001614The cyclic permutation representation number of a skew partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001828The Euler characteristic of a graph. St001850The number of Hecke atoms of a permutation. St001874Lusztig's a-function for the symmetric group. St001933The largest multiplicity of a part in an integer partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000024The number of double up and double down steps of a Dyck path. St000027The major index of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000157The number of descents of a standard tableau. St000169The cocharge of a standard tableau. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000330The (standard) major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000360The number of occurrences of the pattern 32-1. St000376The bounce deficit of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000534The number of 2-rises of a permutation. St000682The Grundy value of Welter's game on a binary word. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000961The shifted major index of a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. 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. St001185The number of indecomposable injective modules of grade at least 2 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 St001340The cardinality of a minimal non-edge isolating set of a graph. St001402The number of separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001413Half the length of the longest even length palindromic prefix 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001524The degree of symmetry of a binary word. St001695The natural comajor index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001902The number of potential covers of a poset. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001956The comajor index for set-valued two-row standard Young tableaux. St000037The sign of a permutation. St000936The number of even values of the symmetric group character corresponding to the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001128The exponens consonantiae of a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000693The modular (standard) major index of a standard tableau. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000642The size of the smallest orbit of antichains under Panyushev complementation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000015The number of peaks of a Dyck path. St000294The number of distinct factors of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000518The number of distinct subsequences in a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000643The size of the largest orbit of antichains under Panyushev complementation. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. 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. 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

$c_{n-1}$ . St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. 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: 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001471The magnitude of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001530The depth of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001720The minimal length of a chain of small intervals in a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001885The number of binary words with the same proper border set. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001846The number of elements which do not have a complement in the lattice. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000144The pyramid weight of the Dyck path. St000289The decimal representation of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 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. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules 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. 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. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001669The number of single rises in a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000759The smallest missing part in an integer partition. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001361The number of lattice paths of the same length that stay weakly above a Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching.