St000522The number of 1-protected nodes of a rooted tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000996The number of exclusive left-to-right maxima of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000386The number of factors DDU in a Dyck path. St000761The number of ascents in an integer composition. St000925The number of topologically connected components of a set partition. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000619The number of cyclic descents of a permutation. St000646The number of big ascents of a permutation. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000647The number of big descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001432The order dimension of the 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. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001118The acyclic chromatic index of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001845The number of join irreducibles minus the rank of a lattice. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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 $eA$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $eA$. St000383The last part of an integer composition. St000657The smallest part of an integer composition. 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 $eA$. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of 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. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group 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. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding 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. 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). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000443The number of long tunnels of a Dyck path. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation 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. 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. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. 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. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. 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. St001884The number of borders of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000331The number of upper interactions of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let 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. 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. 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. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000544The cop number of a graph. St000916The packing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000273The domination number of a graph. St000917The open packing number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. 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 $g$. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St000928The sum of the coefficients of the character polynomial of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000553The number of blocks of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000120The number of left tunnels of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001128The exponens consonantiae of a partition. St000444The length of the maximal rise of a Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001624The breadth of a lattice. St000317The cycle descent number of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of 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. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001626The number of maximal proper sublattices of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001335The cardinality of a minimal cycle-isolating set of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000741The Colin de Verdière graph invariant. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001463The number of distinct columns in the nullspace of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001532The leading coefficient of the Poincare polynomial of the poset cone. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001743The discrepancy of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001792The arboricity of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000455The second largest eigenvalue of a graph if it is integral. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset.