St001458The rank of the adjacency matrix of a graph. St000831The number of indices that are either descents or recoils. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000495The number of inversions of distance at most 2 of a permutation. St000653The last descent of a permutation. St000673The number of non-fixed points of a permutation. St000796The stat' of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000995The largest even part of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000494The number of inversions of distance at most 3 of a permutation. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word. St000877The depth of the binary word interpreted as a path. St000638The number of up-down runs of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001703The villainy of a graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001118The acyclic chromatic index of a graph. St000983The length of the longest alternating subword. St000885The number of critical steps in the Catalan decomposition of a binary word. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001769The reflection length of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000389The number of runs of ones of odd length in a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000762The sum of the positions of the weak records of an integer composition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000455The second largest eigenvalue of a graph if it is integral. St001372The length of a longest cyclic run of ones of a binary word. St000526The number of posets with combinatorially isomorphic order polytopes. St000291The number of descents of a binary word. St001115The number of even descents of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001524The degree of symmetry of a binary word. St000295The length of the border of a binary word. St000352The Elizalde-Pak rank of a permutation. St000524The number of posets with the same order polynomial. St000651The maximal size of a rise in a permutation. St000691The number of changes of a binary word. St000696The number of cycles in the breakpoint graph of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000767The number of runs in an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000903The number of different parts of an integer composition. St001415The length of the longest palindromic prefix of a binary word. St001817The number of flag weak exceedances of a signed permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001488The number of corners of a skew partition. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000525The number of posets with the same zeta polynomial. 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. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000141The maximum drop size of a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001394The genus of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000402Half the size of the symmetry class of a permutation. St000451The length of the longest pattern of the form k 1 2. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000035The number of left outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000012The area of a Dyck path. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000408The number of occurrences of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000670The reversal length of a permutation. 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. St000984The number of boxes below precisely one peak. St001083The number of boxed occurrences of 132 in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000635The number of strictly order preserving maps of a poset into itself. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St000662The staircase size of the code of a permutation. St000288The number of ones in a binary word. St001861The number of Bruhat lower covers of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000871The number of very big ascents of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000058The order of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000883The number of longest increasing subsequences of a permutation. St001058The breadth of the ordered tree. St001371The length of the longest Yamanouchi prefix of a binary word. St001699The major index of a standard tableau minus the weighted size of its shape. St000028The number of stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000456The monochromatic index of a connected graph. St000519The largest length of a factor maximising the subword complexity. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000922The minimal number such that all substrings of this length are unique. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001433The flag major index of a signed permutation. St000260The radius of a connected graph. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001153The number of blocks with even minimum in a set partition. St001267The length of the Lyndon factorization of the binary word. St001669The number of single rises in a Dyck path. St000509The diagonal index (content) of a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. 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. 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. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000401The size of the symmetry class of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000089The absolute variation of a composition. St000173The segment statistic of a semistandard tableau. St000248The number of anti-singletons of a set partition. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000491The number of inversions of a set partition. St000502The number of successions of a set partitions. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001160The number of proper blocks (or intervals) of a permutations. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001727The number of invisible inversions of a permutation. St001843The Z-index of a set partition. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000172The Grundy number of a graph. St000221The number of strong fixed points of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000264The girth of a graph, which is not a tree. 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. St000364The exponent of the automorphism group of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000675The number of centered multitunnels of a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000822The Hadwiger number of the graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000908The length of the shortest maximal antichain in a poset. 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 $c_{n-1}$. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001963The tree-depth of a graph. St000007The number of saliances of the permutation. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000056The decomposition (or block) number of a permutation. St000060The greater neighbor of the maximum. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000310The minimal degree of a vertex of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000354The number of recoils of a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. St000490The intertwining number of a set partition. St000503The maximal difference between two elements in a common block. St000527The width of the poset. St000536The pathwidth of a graph. St000570The Edelman-Greene number of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000730The maximal arc length of a set partition. St000740The last entry of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000886The number of permutations with the same antidiagonal sums. St000909The number of maximal chains of maximal size in a poset. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000981The length of the longest zigzag subpath. St000988The orbit size of a permutation under Foata's bijection. St000996The number of exclusive left-to-right maxima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. 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. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001081The number of minimal length factorizations of a permutation into star transpositions. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001119The length of a shortest maximal path in a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. 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 $K[x]/(x^n)$. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. 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. St001246The maximal difference between two consecutive entries of a permutation. St001270The bandwidth of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001277The degeneracy of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001346The number of parking functions that give the same permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001468The smallest fixpoint of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001498The normalised height of a Nakayama algebra with magnitude 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001625The Möbius invariant of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001694The number of maximal dissociation sets in a graph. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001792The arboricity of a graph. St001806The upper middle entry of a permutation. St001850The number of Hecke atoms of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001928The number of non-overlapping descents in a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St000004The major index of a permutation. St000023The number of inner peaks 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. St000055The inversion sum of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000091The descent variation of a composition. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000216The absolute length of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000247The number of singleton blocks of a set partition. St000304The load of a permutation. St000309The number of vertices with even degree. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) 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. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000680The Grundy value for Hackendot on posets. St000682The Grundy value of Welter's game on a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000879The number of long braid edges in the graph of braid moves of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001073The number of nowhere zero 3-flows of a graph. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. 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. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001306The number of induced paths on four vertices in a graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001377The major index minus the number of inversions of a permutation. St001388The number of non-attacking neighbors of a permutation. St001402The number of separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001434The number of negative sum pairs of a signed permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001726The number of visible inversions of a permutation. St001728The number of invisible descents of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 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. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001781The interlacing number of a set partition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001911A descent variant minus the number of inversions. St000037The sign of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001587Half of the largest even part of an integer partition. St000251The number of nonsingleton blocks of a set partition. St000311The number of vertices of odd degree in a graph. St000694The number of affine bounded permutations that project to a given permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St001623The number of doubly irreducible elements of a lattice. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001720The minimal length of a chain of small intervals in a lattice. 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. St000259The diameter of a connected graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000889The number of alternating sign matrices with the same antidiagonal sums. St001260The permanent of an alternating sign matrix. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001926Sparre Andersen's position of the maximum of a signed permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000550The number of modular elements of a lattice. St000632The jump number of the poset. St000863The length of the first row of the shifted shape of a permutation. St000894The trace of an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001520The number of strict 3-descents. St001570The minimal number of edges to add to make a graph Hamiltonian. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001846The number of elements which do not have a complement in the lattice. St000806The semiperimeter of the associated bargraph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts.