St000013The height of a Dyck path. St000080The rank of the poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. 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: St000528The height of a poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000141The maximum drop size of a permutation. St000317The cycle descent number of a permutation. St000662The staircase size of the code of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001068Number of torsionless simple modules 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001820The size of the image of the pop stack sorting operator. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001626The number of maximal proper sublattices of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. 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. St000340The number of non-final maximal constant sub-paths of length greater than one. St000358The number of occurrences of the pattern 31-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000802The number of occurrences of the vincular pattern |321 in 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. St000864The number of circled entries of the shifted recording tableau of a permutation. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001277The degeneracy of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001358The largest degree of a regular subgraph of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001489The maximum of the number of descents and the number of inverse descents. 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. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001578The minimal number of edges to add or remove to make a graph a line graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001651The Frankl number of a lattice. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000007The number of saliances of the permutation. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000354The number of recoils of a permutation. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000702The number of weak deficiencies of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000877The depth of the binary word interpreted as a path. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001235The global 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001323The independence gap of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001590The crossing number of a perfect matching. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001963The tree-depth of a graph. St000382The first part of an integer composition. St000439The position of the first down step of a Dyck path. St000734The last entry in the first row of a standard tableau. St000907The number of maximal antichains of minimal length in a poset. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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 $c_{n-1}$. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000297The number of leading ones in a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. 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. St000925The number of topologically connected components of a set partition. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000143The largest repeated part of a partition. St000618The number of self-evacuating tableaux of given shape. St001372The length of a longest cyclic run of ones of a binary word. St001424The number of distinct squares in a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000455The second largest eigenvalue of a graph if it is integral. St000366The number of double descents of a permutation. St000871The number of very big ascents of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000035The number of left outer peaks of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000023The number of inner peaks of a permutation. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St000456The monochromatic index of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000254The nesting number of a set partition. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001737The number of descents of type 2 in a permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000824The sum of the number of descents and the number of recoils of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees 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. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000422The energy of a graph, if it is integral. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000736The last entry in the first row of a semistandard tableau. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion.