St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St000667The greatest common divisor of the parts of the partition. St000897The number of different multiplicities of parts of an integer partition. St001881The number of factors of a lattice as a Cartesian product of lattices. St001933The largest multiplicity of a part in an integer partition. 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. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000296The length of the symmetric border of a binary word. St000377The dinv defect of an integer partition. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St000921The number of internal inversions of a binary word. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000995The largest even part of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001248Sum of the even parts of a partition. St001252Half the sum of the even parts of a partition. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001696The natural major index of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001961The sum of the greatest common divisors of all pairs of parts. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. 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. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000382The first part of an integer composition. St000392The length of the longest run of ones in a binary word. 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. St000548The number of different non-empty partial sums of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St000805The number of peaks of the associated bargraph. 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. 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. St000816The number of standard composition tableaux of the composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of 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. 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. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. 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. 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 $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001256Number of simple reflexive modules that are 2-stable reflexive. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given 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. St001481The minimal height of a peak of a Dyck path. St001484The number of singletons of an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000630The length of the shortest palindromic decomposition of a binary word. St000759The smallest missing part in an integer partition. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. 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. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000284The Plancherel distribution on integer partitions. St000659The number of rises of length at least 2 of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001128The exponens consonantiae of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle 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. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. 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. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001561The value of the elementary symmetric function evaluated at 1. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000363The number of minimal vertex covers of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000090The variation of a composition. St000268The number of strongly connected orientations of a graph. St000313The number of degree 2 vertices of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000637The length of the longest cycle in a graph. St001073The number of nowhere zero 3-flows of a graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001374The Padmakar-Ivan index of a graph. St001396Number of triples of incomparable elements in a finite poset. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001638The book thickness of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001689The number of celebrities in a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001736The total number of cycles in a graph. St001764The number of non-convex subsets of vertices in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000260The radius of a connected graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000273The domination number of a graph. St000315The number of isolated vertices of a graph. St000383The last part of an integer composition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000916The packing number of a graph. St000948The chromatic discriminant of a graph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001272The number of graphs with the same degree sequence. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number 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. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001546The number of monomials in the Tutte polynomial of a graph. 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. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001691The number of kings in a graph. St001694The number of maximal dissociation sets in a graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001739The number of graphs with the same edge polytope as the given 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. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000258The burning number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000914The sum of the values of the Möbius function of a poset. St000633The size of the automorphism group of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000699The toughness times the least common multiple of 1,. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001281The normalized isoperimetric number of a graph. 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. St000455The second largest eigenvalue of a graph if it is integral. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001846The number of elements which do not have a complement in the lattice. St000456The monochromatic index of a connected graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001820The size of the image of the pop stack sorting operator. St001890The maximum magnitude of the Möbius function of a poset. St001060The distinguishing index of a graph. 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 $eA$. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001765The number of connected components of the friends and strangers graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000311The number of vertices of odd degree in a graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type.