St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000764The number of strong records in an integer composition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001730The number of times the path corresponding to a binary word crosses the base line. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000544The cop number of a graph. St000664The number of right ropes of a permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000386The number of factors DDU in a Dyck path. St000665The number of rafts of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000225Difference between largest and smallest parts in a partition. St000944The 3-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000872The number of very big descents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St001728The number of invisible descents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000260The radius of a connected graph. St000353The number of inner valleys 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000929The constant term of the character polynomial of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000650The number of 3-rises of a permutation. 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. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001399The distinguishing number of a poset. St000658The number of rises of length 2 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001651The Frankl number of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001513The number of nested exceedences of a permutation. St000633The size of the automorphism group of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001820The size of the image of the pop stack sorting operator. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension 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. St000731The number of double exceedences of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000022The number of fixed points of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001624The breadth of a lattice. St001403The number of vertical separators in a permutation. St001470The cyclic holeyness of a permutation. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000407The number of occurrences of the pattern 2143 in a permutation. St000649The number of 3-excedences of a permutation. St000732The number of double deficiencies of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000570The Edelman-Greene number of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000527The width of the poset. St000137The Grundy value of an integer partition. St000618The number of self-evacuating tableaux of given shape. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001301The first Betti number of the order complex associated with the poset. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St000374The number of exclusive right-to-left minima of a permutation. St000908The length of the shortest maximal antichain in a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001720The minimal length of a chain of small intervals in a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000627The exponent of a binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000781The number of proper colouring schemes of a Ferrers diagram. St001383The BG-rank of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001964The interval resolution global dimension of a poset. St000284The Plancherel distribution on integer partitions. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. 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. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. 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. 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. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000181The number of connected components of the Hasse diagram for the poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001811The Castelnuovo-Mumford regularity of a permutation. St001846The number of elements which do not have a complement in the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000741The Colin de Verdière graph invariant. St001397Number of pairs of incomparable elements in a finite poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000058The order of a permutation. St000768The number of peaks in an integer composition. St000352The Elizalde-Pak rank of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000989The number of final rises of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001487The number of inner corners of a skew partition. 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. St000237The number of small exceedances. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000920The logarithmic height of a Dyck path. St000666The number of right tethers of a permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001665The number of pure excedances of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000516The number of stretching pairs of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001061The number of indices that are both descents and recoils of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001948The number of augmented double ascents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000805The number of peaks of the associated bargraph. St000877The depth of the binary word interpreted as a path. St000899The maximal number of repetitions of an integer composition. St000991The number of right-to-left minima of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000392The length of the longest run of ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. 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 $eA$. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000883The number of longest increasing subsequences of a permutation.