St000759The smallest missing part in an integer partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000932The number of occurrences of the pattern UDU in a 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000146The Andrews-Garvan crank of a partition. St000678The number of up steps after the last double rise of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001885The number of binary words with the same proper border set. 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 $c_{n-1}$. St000024The number of double up and double down steps of a Dyck path. St000153The number of adjacent cycles of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000288The number of ones in a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000409The number of pitchforks in a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000946The sum of the skew hook positions in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000976The sum of the positions of double up-steps of a Dyck path. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. 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. St001424The number of distinct squares in a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001510The number of self-evacuating linear extensions of a finite poset. St001524The degree of symmetry of a binary word. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001721The degree of a binary word. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. 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). St001931The weak major index of an integer composition regarded as a word. St001948The number of augmented double ascents of a permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000680The Grundy value for Hackendot on posets. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St000990The first ascent of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. 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. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001377The major index minus the number of inversions of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001481The minimal height of a peak of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001884The number of borders of a binary word. St000439The position of the first down step of a Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000981The length of the longest zigzag subpath. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000008The major index of the composition. St000012The area of a Dyck path. 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. St000035The number of left outer peaks of a permutation. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000057The Shynar inversion number of a standard tableau. St000083The number of left oriented leafs of a binary tree except the first one. St000088The row sums of the character table of the symmetric group. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000096The number of spanning trees of a graph. St000120The number of left tunnels of a Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000133The "bounce" of a permutation. St000137The Grundy value of an integer partition. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000154The sum of the descent bottoms 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. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. 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. St000225Difference between largest and smallest parts in a partition. 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. St000245The number of ascents of a permutation. St000247The number of singleton blocks 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. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000268The number of strongly connected orientations of a graph. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000374The number of exclusive right-to-left minima of a permutation. St000376The bounce deficit of a Dyck path. 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. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000441The number of successions of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000472The sum of the ascent bottoms of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000498The lcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000546The number of global descents of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000624The normalized sum of the minimal distances to a greater element. St000628The balance 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. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. 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. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. 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. St000850The number of 1/2-balanced pairs in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. 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. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St000947The major index east count of a Dyck path. St000948The chromatic discriminant of a graph. St000951The dimension of $Ext^{1}(D(A),A)$ 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. 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. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001057The Grundy value of the game of creating an independent set in a graph. St001061The number of indices that are both descents and recoils of a permutation. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001094The depth index of a set partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular 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. St001161The major index north count of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001234The number of indecomposable three 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001260The permanent of an alternating sign matrix. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. 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. St001281The normalized isoperimetric number of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001353The number of prime nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001367The smallest number which does not occur as degree of a vertex in a graph. St001375The pancake length of a permutation. St001381The fertility of a permutation. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001405The number of bonds in a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001423The number of distinct cubes 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001479The number of bridges of a graph. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001552The number of inversions between excedances and fixed points of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001561The value of the elementary symmetric function evaluated at 1. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001657The number of twos in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001675The number of parts equal to the part in the reversed composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. 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. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001781The interlacing number of a set partition. St001783The number of odd automorphisms of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001826The maximal number of leaves on a vertex of a graph. St001850The number of Hecke atoms 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001928The number of non-overlapping descents in a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001957The number of Hasse diagrams with a given underlying undirected graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. 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. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000082The number of elements smaller than a binary tree in Tamari order. St000084The number of subtrees. St000099The number of valleys of a permutation, including the boundary. St000145The Dyson rank of a partition. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000166The depth minus 1 of an ordered tree. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000240The number of indices that are not small excedances. St000255The number of reduced Kogan faces with the permutation as type. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000307The number of rowmotion orbits of a poset. 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. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000351The determinant of the adjacency matrix of a graph. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000402Half the size of the symmetry class of a permutation. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000417The size of the automorphism group of the ordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000478Another weight of a partition according to Alladi. 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. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000531The leading coefficient of the rook polynomial of an integer partition. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000633The size of the automorphism group of a poset. St000638The number of up-down runs of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000657The smallest part of an integer composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000700The protection number of an ordered tree. St000702The number of weak deficiencies of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000717The number of ordinal summands of a poset. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. 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. St000823The number of unsplittable factors of the set partition. St000831The number of indices that are either descents or recoils. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. 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. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. 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. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000906The length of the shortest maximal chain in a poset. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St000917The open packing number of a graph. St000920The logarithmic height of a Dyck path. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. 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 $c_{n−1}$. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St000991The number of right-to-left minima of a permutation. St000993The multiplicity of the largest part of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in 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. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001096The size of the overlap set of a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001128The exponens consonantiae of a partition. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama 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: St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001316The domatic number of a graph. 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. St001368The number of vertices of maximal degree in a graph. St001399The distinguishing number of a poset. St001459The number of zero columns in the nullspace of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001486The number of corners of the ribbon associated with an integer composition. St001497The position of the largest weak excedence of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001516The number of cyclic bonds of a permutation. St001530The depth of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. 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. 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. St001669The number of single rises in a Dyck path. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001778The largest greatest common divisor of an element and its image in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001933The largest multiplicity of a part in an integer partition. St001937The size of the center of a parking function. St001959The product of the heights of the peaks of a Dyck path. St000094The depth of an ordered tree. St000144The pyramid weight of the Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000378The diagonal inversion number of an integer partition. St000521The number of distinct subtrees of an ordered tree. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. 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. St000738The first entry in the last row of a standard tableau. 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. St000891The number of distinct diagonal sums of a permutation matrix. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a permutation. St001376The Colless index of a binary tree. St001468The smallest fixpoint of a permutation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001566The length of the longest arithmetic progression in a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000756The sum of the positions of the left to right maxima of a permutation. St000922The minimal number such that all substrings of this length are unique. St000973The length of the boundary of an ordered tree. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. 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. St001814The number of partitions interlacing the given partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.