St000098The chromatic number of a graph. St000154The sum of the descent bottoms of a permutation. St000172The Grundy number of a graph. St000182The number of permutations whose cycle type is the given integer partition. 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. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000722The number of different neighbourhoods in a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. St000822The Hadwiger number of the graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. 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. 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. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001763The Hurwitz number of an integer partition. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001850The number of Hecke atoms of a permutation. 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. St000290The major index of a binary word. St000293The number of inversions of a binary word. 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. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. 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. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000465The first Zagreb index of a graph. St000534The number of 2-rises of a permutation. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000571The F-index (or forgotten topological index) of a graph. St000682The Grundy value of Welter's game on a binary word. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000769The major index of a composition regarded as a word. St000846The maximal number of elements covering an element of a poset. St000915The Ore degree of a graph. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. 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. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. 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. St001485The modular major index of a binary word. 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. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001869The maximum cut size of a graph. St001902The number of potential covers of a poset. St001962The proper pathwidth of a graph. St001472The permanent of the Coxeter matrix of the poset. St000001The number of reduced words for a permutation. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000078The number of alternating sign matrices whose left key is the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000141The maximum drop size of a permutation. St000163The size of the orbit of the set partition under rotation. St000238The number of indices that are not small weak excedances. St000255The number of reduced Kogan faces with the permutation as type. St000273The domination number of a graph. St000287The number of connected components of a graph. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000416The number of inequivalent increasing trees of an ordered tree. St000472The sum of the ascent bottoms of a permutation. St000529The number of permutations whose descent word is the given binary word. St000539The number of odd inversions of a permutation. St000543The size of the conjugacy class of a binary word. 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. St000626The minimal period of a binary word. St000627The exponent of a binary word. 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. St000738The first entry in the last row of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000756The sum of the positions of the left to right maxima of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000765The number of weak records in an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000792The Grundy value for the game of ruler on a binary word. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000847The number of standard Young tableaux whose descent set is the binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. 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. St000946The sum of the skew hook positions in a Dyck path. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001363The Euler characteristic of a graph according to Knill. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. 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. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001528The number of permutations such that the product with the permutation has the same number of fixed points. 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. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001778The largest greatest common divisor of an element and its image in a permutation. 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. St001813The product of the sizes of the principal order filters in a poset. St001829The common independence number of a graph. St001838The number of nonempty primitive factors of a binary word. St001851The number of Hecke atoms of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000004The major index of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000027The major index of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000091The descent variation of a composition. 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. St000156The Denert index of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000295The length of the border of a binary word. St000305The inverse major index of a permutation. St000330The (standard) major index of a standard tableau. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000461The rix statistic of a permutation. St000484The sum of St000483 over all subsequences of length at least three. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000538The number of even inversions of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000638The number of up-down runs of a permutation. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000673The number of non-fixed points of a permutation. St000676The number of odd rises of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000766The number of inversions of an integer composition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St000824The sum of the number of descents and the number of recoils of a permutation. St000828The spearman's rho of a permutation and the identity permutation. St000836The number of descents of distance 2 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000933The number of multipartitions of sizes given by an integer partition. St000963The 2-shifted major index of a permutation. St000979Half of MacMahon's equal index of a Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001058The breadth of the ordered tree. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001130The number of two successive successions in a permutation. St001160The number of proper blocks (or intervals) of a permutations. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001248Sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001308The number of induced paths on three vertices in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001350Half of the Albertson index of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001402The number of separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001423The number of distinct cubes in a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001524The degree of symmetry of a binary word. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001671Haglund's hag of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001696The natural major index of a standard Young tableau. St001705The number of occurrences of the pattern 2413 in a permutation. St001721The degree of a binary word. St001727The number of invisible inversions of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001759The Rajchgot index of a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001807The lower middle entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001893The flag descent of a signed permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St000037The sign of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000936The number of even values of the symmetric group character corresponding to the partition. St000961The shifted major index of a permutation. St000690The size of the conjugacy class of a permutation. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000219The number of occurrences of the pattern 231 in a permutation. St000653The last descent of a permutation. St000693The modular (standard) major index of a standard tableau. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000830The total displacement of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000181The number of connected components of the Hasse diagram for the poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000284The Plancherel distribution on integer partitions. St000388The number of orbits of vertices of a graph under automorphisms. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001057The Grundy value of the game of creating an independent set in a graph. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001389The number of partitions of the same length below the given integer partition. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001527The cyclic permutation representation number of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001780The order of promotion on the set of standard tableaux of given shape. 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. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. 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. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000060The greater neighbor of the maximum. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000100The number of linear extensions of a poset. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000266The number of spanning subgraphs of a graph with the same connected components. St000268The number of strongly connected orientations of a graph. 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. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. 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. St000343The number of spanning subgraphs of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000347The inversion sum of a binary word. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000509The diagonal index (content) of a partition. St000530The number of permutations with the same descent word as the given permutation. St000531The leading coefficient of the rook polynomial of an integer partition. 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. St000568The hook number of a binary tree. St000570The Edelman-Greene number of a permutation. St000619The number of cyclic descents of a permutation. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. 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. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000734The last entry in the first row of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000782The indicator function of whether a given perfect matching is an L & P matching. St000783The side length of the largest staircase partition fitting into a partition. 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. 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. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000910The number of maximal chains of minimal length 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. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000947The major index east count of a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000972The composition number of a graph. St000988The orbit size of a permutation under Foata's bijection. St000993The multiplicity of the largest part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. 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. St001024Maximum of dominant dimensions of the simple modules 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. St001052The length of the exterior of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a 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. St001162The minimum jump of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001182Number of indecomposable injective modules with codominant dimension at least two 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 $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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: 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. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001239The largest vector space dimension of the double dual of a simple module in 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001246The maximal difference between two consecutive entries of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001367The smallest number which does not occur as degree of a vertex in a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. 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 $eA$. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000744The length of the path to the largest entry in a standard Young tableau. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.