St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000306The bounce count of a Dyck path. St000327The number of cover relations in a poset. St000331The number of upper interactions of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant 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 $eA$. 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. 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. 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001637The number of (upper) dissectors of a poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001668The number of points of the poset minus the width of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St001955The number of natural descents for set-valued two row standard Young tableaux. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000171The degree of the graph. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length 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. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000336The leg major index of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000445The number of rises of length 1 of a Dyck path. 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. St000528The height of a poset. St000617The number of global maxima of a Dyck path. St000636The hull number of a graph. St000674The number of hills of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000680The Grundy value for Hackendot on posets. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000717The number of ordinal summands of a poset. St000809The reduced reflection length of the permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000890The number of nonzero entries in an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St000924The number of topologically connected components of a perfect matching. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000957The number of Bruhat lower covers of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules 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. 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: St001211The number of simple modules in the corresponding Nakayama algebra that have 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. 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. St001253The number of non-projective indecomposable reflexive modules in 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001437The flex of a binary word. St001480The number of simple summands of the module J^2/J^3. 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. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001958The degree of the polynomial interpolating the values of a permutation. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000673The number of non-fixed points of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000806The semiperimeter of the associated bargraph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001468The smallest fixpoint of a permutation. St000060The greater neighbor of the maximum. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000296The length of the symmetric border of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000553The number of blocks of a graph. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000829The Ulam distance of a permutation to the identity permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000922The minimal number such that all substrings of this length are unique. St000956The maximal displacement of a permutation. St000982The length of the longest constant subword. St001176The size of a partition minus its first part. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001391The disjunction number of a graph. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001727The number of invisible inversions of a permutation. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St000004The major index of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000030The sum of the descent differences of a permutations. St000031The number of cycles in the cycle decomposition of a permutation. St000050The depth or height of a binary tree. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000242The number of indices that are not cyclical small weak excedances. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000358The number of occurrences of the pattern 31-2. St000374The number of exclusive right-to-left minima of a permutation. St000503The maximal difference between two elements in a common block. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000625The sum of the minimal distances to a greater element. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. 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. St000733The row containing the largest entry of a standard tableau. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. 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. St001052The length of the exterior of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001096The size of the overlap set of a permutation. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001430The number of positive entries in a signed permutation. St001554The number of distinct nonempty subtrees of a binary tree. St001622The number of join-irreducible elements of a lattice. St001672The restrained domination number of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001759The Rajchgot index of a permutation. St000054The first entry of the permutation. St000167The number of leaves of an ordered tree. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000451The length of the longest pattern of the form k 1 2. St000505The biggest entry in the block containing the 1. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000867The sum of the hook lengths in the first row of an integer partition. 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. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000719The number of alignments in a perfect matching. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001118The acyclic chromatic index of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001927Sparre Andersen's number of positives of a signed permutation. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001555The order of a signed permutation. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000245The number of ascents of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000338The number of pixed points of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000989The number of final rises of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000461The rix statistic 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. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000654The first descent of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001220The width of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001517The length of a longest pair of twins in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001894The depth of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000782The indicator function of whether a given perfect matching is an L & P matching. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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 $K[x]/(x^n)$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.