St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. 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. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000171The degree of the graph. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001391The disjunction number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001758The number of orbits of promotion on a graph. St001869The maximum cut size of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001725The harmonious chromatic number of a graph. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000088The row sums of the character table of the symmetric group. St000137The Grundy value of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000377The dinv defect of an integer partition. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000535The rank-width of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001271The competition number of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001638The book thickness 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. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000013The height of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000293The number of inversions of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000335The difference of lower and upper interactions. St000378The diagonal inversion number 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. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000482The (zero)-forcing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000537The cutwidth of a graph. St000549The number of odd partial sums of an integer partition. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000681The Grundy value of Chomp on Ferrers diagrams. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000922The minimal number such that all substrings of this length are unique. St000952Gives the number of irreducible factors 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. St000982The length of the longest constant subword. 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. St001034The area of the parallelogram polyomino associated with the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001267The length of the Lyndon factorization of the binary word. St001270The bandwidth of a graph. St001281The normalized isoperimetric number of a graph. St001286The annihilation number of a graph. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001488The number of corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. 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. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. St001742The difference of the maximal and the minimal degree in a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St001962The proper pathwidth of a graph. St000014The number of parking functions supported by a Dyck path. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000271The chromatic index of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000935The number of ordered refinements 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. St001012Number of simple modules with projective dimension at most 2 in 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001112The 3-weak dynamic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. 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 St001389The number of partitions of the same length below the given integer partition. 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000637The length of the longest cycle in a graph. St000806The semiperimeter of the associated bargraph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001838The number of nonempty primitive factors of a binary word. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001706The number of closed sets in a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000009The charge of a standard tableau. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000035The number of left outer peaks of a permutation. St000039The number of crossings of a permutation. St000041The number of nestings of a perfect matching. St000053The number of valleys of the Dyck path. St000055The inversion sum of a permutation. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000089The absolute variation of a composition. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000148The number of odd parts of a partition. St000152The number of boxed plus the number of special entries. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000216The absolute length of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000341The non-inversion sum of a permutation. St000354The number of recoils of a permutation. St000356The number of occurrences of the pattern 13-2. St000374The number of exclusive right-to-left minima of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000398The sum of the depths of the vertices (or total internal path length) of a binary tree. St000409The number of pitchforks in a binary tree. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000486The number of cycles of length at least 3 of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000731The number of double exceedences of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000754The Grundy value for the game of removing nestings in a perfect matching. St000779The tier of a permutation. St000795The mad of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000809The reduced reflection length of the permutation. St000819The propagating number of a perfect matching. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001096The size of the overlap set of a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001160The number of proper blocks (or intervals) of a permutations. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. 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. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001372The length of a longest cyclic run of ones of a binary word. St001376The Colless index of a binary tree. St001388The number of non-attacking neighbors of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001405The number of bonds in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001517The length of a longest pair of twins in a permutation. St001524The degree of symmetry of a binary word. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001673The degree of asymmetry of an integer composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001721The degree of a binary word. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001911A descent variant minus the number of inversions. St001925The minimal number of zeros in a row of an alternating sign matrix. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001958The degree of the polynomial interpolating the values of a permutation. St000001The number of reduced words for a permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000040The number of regions of the inversion arrangement of a permutation. St000050The depth or height of a binary tree. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000134The size of the orbit of an alternating sign matrix under gyration. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000203The number of external nodes of a binary tree. St000210Minimum over maximum difference of elements in cycles. St000213The number of weak exceedances (also weak excedences) of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000240The number of indices that are not small excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000336The leg major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000385The number of vertices with out-degree 1 in a binary tree. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000413The number of ordered trees with the same underlying unordered tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000458The number of permutations obtained by switching adjacencies or successions. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000507The number of ascents of a standard tableau. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000530The number of permutations with the same descent word as the given permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000657The smallest part of an integer composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000676The number of odd rises 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. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000690The size of the conjugacy class of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000702The number of weak deficiencies of a permutation. St000719The number of alignments in a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000767The number of runs in an integer composition. St000780The size of the orbit under rotation of a perfect matching. St000789The number of crossing-similar perfect matchings of a perfect matching. St000808The number of up steps of the associated bargraph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000820The number of compositions obtained by rotating the composition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000863The length of the first row of the shifted shape of a permutation. 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. St000883The number of longest increasing subsequences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000891The number of distinct diagonal sums of a permutation matrix. St000903The number of different parts of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000920The logarithmic height of a Dyck path. St000924The number of topologically connected components of a perfect matching. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000945The number of matchings in the dihedral orbit of a perfect matching. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001077The prefix exchange distance of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001220The width of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001343The dimension of the reduced incidence algebra of a poset. St001346The number of parking functions that give the same permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001375The pancake length of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001485The modular major index of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001530The depth of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001589The nesting number of a perfect matching. St001595The number of standard Young tableaux of the skew partition. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001759The Rajchgot index of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001813The product of the sizes of the principal order filters in a poset. St001815The number of order preserving surjections from a poset to a total order. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001884The number of borders of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St000026The position of the first return of a Dyck path. St000044The number of vertices of the unicellular map given by a perfect matching. St000070The number of antichains in a poset. St000085The number of linear extensions of the tree. St000094The depth of an ordered tree. St000104The number of facets in the order polytope of this poset. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000151The number of facets in the chain polytope of the poset. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000294The number of distinct factors of a binary word. St000439The position of the first down step of a Dyck path. St000471The sum of the ascent tops of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000518The number of distinct subsequences in a binary word. St000521The number of distinct subtrees of an ordered tree. St000673The number of non-fixed points of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000824The sum of the number of descents and the number of recoils of a permutation. St000825The sum of the major and the inverse major index of a permutation. St000842The breadth of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. 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 $c_{n-1}$. St000973The length of the boundary of an ordered tree. St000981The length of the longest zigzag subpath. St000988The orbit size of a permutation under Foata's bijection. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000868The aid statistic in the sense of Shareshian-Wachs. 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. 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. St001468The smallest fixpoint of a permutation. St001814The number of partitions interlacing the given partition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.