St000532The total number of rook placements on a Ferrers board. 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. St000926The clique-coclique number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000008The major index of the composition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000185The weighted size of a partition. St000228The size of a partition. St000271The chromatic index of a graph. St000301The number of facets of the stable set polytope of a graph. St000384The maximal part of the shifted composition of an integer partition. St000448The number of pairs of vertices of a graph with distance 2. St000459The hook length of the base cell of a partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. 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. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001664The number of non-isomorphic subposets of a poset. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000172The Grundy number of a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000286The number of connected components of the complement of a graph. St000290The major index of a binary word. St000363The number of minimal vertex covers of a graph. St000456The monochromatic index of a connected graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000867The sum of the hook lengths in the first row of an integer partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001118The acyclic chromatic index of a graph. St001127The sum of the squares of the parts of a partition. St001302The number of minimally 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. St001342The number of vertices in the center of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. 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. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001611The number of multiset partitions such that the multiplicities of elements 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. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001869The maximum cut size of a graph. St001883The mutual visibility number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001963The tree-depth of a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000160The multiplicity of the smallest part of a partition. St000169The cocharge of a standard tableau. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000445The number of rises of length 1 of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000507The number of ascents of a standard tableau. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000676The number of odd rises of a Dyck path. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001034The area of the parallelogram polyomino associated with the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001161The major index north count of a Dyck path. St001176The size of a partition minus its first part. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001308The number of induced paths on three vertices in a graph. St001350Half of the Albertson index of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001521Half the total irregularity of a graph. St001541The Gini index of an integer partition. St001644The dimension of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001697The shifted natural comajor index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001721The degree of a binary word. St001783The number of odd automorphisms of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001812The biclique partition number of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001949The rigidity index of a graph. St001956The comajor index for set-valued two-row standard Young tableaux. St001962The proper pathwidth of a graph. St000004The major index of a permutation. St000020The rank of the permutation. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000446The disorder of a permutation. St000631The number of distinct palindromic decompositions of a binary word. St000794The mak of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St000009The charge of a standard tableau. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000289The decimal representation of a binary word. St000304The load of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000868The aid statistic in the sense of Shareshian-Wachs. St000947The major index east count of a Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001362The normalized Knill dimension of a graph. St001375The pancake length of a permutation. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001658The total number of rook placements on a Ferrers board. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000668The least common multiple of the parts of the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001500The global dimension of magnitude 1 Nakayama algebras. St001959The product of the heights of the peaks of a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001808The box weight or horizontal decoration of a Dyck path. St000259The diameter of a connected graph. St000472The sum of the ascent bottoms of a permutation. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St001077The prefix exchange distance of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000914The sum of the values of the Möbius function of a poset. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. 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. St000048The multinomial of the parts of a partition. St000088The row sums of the character table of the symmetric group. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000293The number of inversions of a binary word. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000519The largest length of a factor maximising the subword complexity. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000544The cop number of a graph. St000553The number of blocks of a graph. St000567The sum of the products of all pairs of parts. St000626The minimal period of a binary word. 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. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. 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. 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. St000910The number of maximal chains of minimal length in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. 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. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. 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 $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000982The length of the longest constant subword. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000993The multiplicity of the largest part of an integer partition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one 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. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001267The length of the Lyndon factorization of the binary word. St001281The normalized isoperimetric number of a graph. St001286The annihilation number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. 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. 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. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001378The product of the cohook lengths of the integer partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001561The value of the elementary symmetric function evaluated at 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001614The cyclic permutation representation number of a skew partition. St001617The dimension of the space of valuations of a lattice. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001691The number of kings in a graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001733The number of weak left to right maxima of a Dyck path. St001757The number of orbits of toric promotion on a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001834The number of non-isomorphic minors of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St000422The energy of a graph, if it is integral. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000038The product of the heights of the descending steps of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000477The weight of a partition according to Alladi. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000808The number of up steps of the associated bargraph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000981The length of the longest zigzag subpath. St000983The length of the longest alternating subword. 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. St001008Number of indecomposable injective 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. St001012Number of simple modules with projective dimension at most 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in 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. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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 $g$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001313The number of Dyck paths above the lattice path given by a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of 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. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001485The modular major index of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001885The number of binary words with the same proper border set. St001910The height of the middle non-run of a Dyck path. St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000699The toughness times the least common multiple of 1,. St001060The distinguishing index of a graph. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000455The second largest eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000294The number of distinct factors of a binary word. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000346The number of coarsenings of a partition. St000347The inversion sum of a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000518The number of distinct subsequences in a binary word. St000627The exponent of a binary word. St000644The number of graphs with given frequency partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000762The sum of the positions of the weak records of an integer composition. 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. St000869The sum of the hook lengths of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. 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. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001129The product of the squares of the parts of a partition. St001199The dominant dimension of $eAe$ for 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 $eA$. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001361The number of lattice paths of the same length that stay weakly above a Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001531Number of partial orders contained in the poset determined by the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001732The number of peaks visible from the left. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire.