St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000214The number of adjacencies of a permutation. St000247The number of singleton blocks of a set partition. St000441The number of successions of a permutation. St000502The number of successions of a set partitions. St000658The number of rises of length 2 of a Dyck path. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St000906The length of the shortest maximal chain in a poset. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001399The distinguishing number of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000444The length of the maximal rise of a Dyck path. St001115The number of even descents of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St000989The number of final rises of a permutation. St001948The number of augmented double ascents of a permutation. St000338The number of pixed points of a permutation. St000732The number of double deficiencies of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000990The first ascent of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. 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. 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000241The number of cyclical small excedances. St000291The number of descents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000389The number of runs of ones of odd length in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000819The propagating number of a perfect matching. St000836The number of descents of distance 2 of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000940The number of characters of the symmetric group whose value on the partition is zero. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001114The number of odd descents of 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001910The height of the middle non-run of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St001423The number of distinct cubes in a binary word. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000454The largest eigenvalue of a graph if it is integral. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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 $eA$. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000933The number of multipartitions of sizes given by an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000659The number of rises of length at least 2 of a Dyck path. 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. St000946The sum of the skew hook positions in a Dyck path. St000993The multiplicity of the largest part of an integer partition. 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St000045The number of linear extensions of a binary tree. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000326The position of the first one in a binary word after appending a 1 at the end. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000478Another weight of a partition according to Alladi. St000487The length of the shortest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000504The cardinality of the first block of a set partition. St000539The number of odd inversions 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. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000653The last descent of a permutation. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000886The number of permutations with the same antidiagonal sums. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000919The number of maximal left branches of a binary tree. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000988The orbit size of a permutation under Foata's bijection. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001346The number of parking functions that give the same permutation. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001959The product of the heights of the peaks of a Dyck path. St000137The Grundy value of an integer 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. 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. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000443The number of long tunnels of a Dyck path. St000475The number of parts equal to 1 in a partition. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000549The number of odd partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. 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. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. 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 $c_{n−1}$. St000983The length of the longest alternating subword. St000992The alternating sum of the parts of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001129The product of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the 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: St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001224Let X be the direct sum of all simple modules of 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle 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. St001612The number of coloured multisets of cycles 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. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000456The monochromatic index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000567The sum of the products of all pairs of parts. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001118The acyclic chromatic index of a graph. St001176The size of a partition minus its first part. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000260The radius of a connected graph. St001096The size of the overlap set of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000284The Plancherel distribution on integer partitions. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000663The number of right floats of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. 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. 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. St001153The number of blocks with even minimum in a set partition. 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 $eA$. St000090The variation of a composition. St000455The second largest eigenvalue of a graph if it is integral. St001050The number of terminal closers of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000477The weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. 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. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000928The sum of the coefficients 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. St000997The even-odd crank of an integer partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001808The box weight or horizontal decoration of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone.