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. 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. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. 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. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St000983The length of the longest alternating subword. 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. St001500The global dimension of magnitude 1 Nakayama algebras. 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. St001620The number of sublattices of a lattice. 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. St001706The number of closed sets in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001757The number of orbits of toric promotion on a graph. St001762The number of convex subsets of vertices 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. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a 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. 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. 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. 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 $K[x]/(x^n)$. 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. 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. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. 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. St000109The number of elements less than or equal to the given element in Bruhat order. 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. St000418The number of Dyck paths that are weakly below a Dyck path. 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. St000644The number of graphs with given frequency partition. 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. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of 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. 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. St000961The shifted major index of a permutation. St000982The length of the longest constant subword. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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 St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. 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. St001524The degree of symmetry of a binary word. 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. 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. St001959The product of the heights of the peaks of a Dyck path. St000037The sign of a permutation. St000519The largest length of a factor maximising the subword complexity. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001486The number of corners of the ribbon associated with an integer composition. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001057The Grundy value of the game of creating an independent set in a graph. St001118The acyclic chromatic index of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 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. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001890The maximum magnitude of the Möbius function of a poset. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000005The bounce statistic of a Dyck path. St000015The number of peaks 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. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000108The number of partitions contained in the given partition. St000120The number of left tunnels of a Dyck path. St000137The Grundy value of an integer partition. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000266The number of spanning subgraphs of a graph with the same connected components. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284The Plancherel distribution on integer partitions. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000291The number of descents of a binary word. St000296The length of the symmetric border of a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000379The number of Hamiltonian cycles in a graph. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000509The diagonal index (content) of a partition. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000708The product of the parts of an integer partition. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000717The number of ordinal summands of a poset. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000770The major index of an integer partition when read from bottom to top. St000783The side length of the largest staircase partition fitting into a partition. St000792The Grundy value for the game of ruler on a binary word. St000805The number of peaks of the associated bargraph. 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000816The number of standard composition tableaux of the composition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000820The number of compositions obtained by rotating the composition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000906The length of the shortest maximal chain in a poset. St000913The number of ways to refine the partition into singletons. St000914The sum of the values of the Möbius function of a poset. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000947The major index east count of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000972The composition number of a graph. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001095The number of non-isomorphic posets with precisely one further covering relation. St001109The number of proper colourings of a graph with as few colours as possible. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. 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. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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: St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001256Number of simple reflexive modules that are 2-stable reflexive. St001261The Castelnuovo-Mumford regularity of a graph. St001267The length of the Lyndon factorization of the binary word. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001360The number of covering relations in Young's lattice below a partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001477The number of nowhere zero 5-flows of a graph. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001530The depth of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001820The size of the image of the pop stack sorting operator. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001955The number of natural descents for set-valued two row standard Young tableaux. 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. St000006The dinv of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000244The cardinality of the automorphism group of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000347The inversion sum of a binary word. St000364The exponent of the automorphism group of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. 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. St000567The sum of the products of all pairs of parts. 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. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000707The product of the factorials of the parts. St000735The last entry on the main diagonal of a standard tableau. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000948The chromatic discriminant of a graph. 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 $c_{n−1}$. 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. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. 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. 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. St001123The multiplicity of the dual 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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). St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms 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). St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001877Number of indecomposable injective modules with projective dimension 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. 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. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001930The weak major index of a binary word.