St001328The minimal number of occurrences of the bipartite-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. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001350Half of the Albertson index of a graph. 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. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001736The total number of cycles in a graph. St001764The number of non-convex subsets of vertices in a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000080The rank of the poset. St000081The number of edges of a graph. St000091The descent variation of a composition. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000171The degree of the graph. 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. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000313The number of degree 2 vertices of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000845The maximal number of elements covered by an element in a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001175The size of a partition minus the hook length of the base cell. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. 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. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series 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. St001362The normalized Knill dimension of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001396Number of triples of incomparable elements in a finite poset. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001521Half the total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001962The proper pathwidth of a graph. St000003The number of standard Young tableaux of the partition. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000086The number of subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000268The number of strongly connected orientations of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers 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. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000482The (zero)-forcing number of a graph. St000528The height of a poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000722The number of different neighbourhoods in a graph. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. 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. St000822The Hadwiger number of the graph. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000948The chromatic discriminant of a graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. 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. St001432The order dimension of the partition. St001463The number of distinct columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001494The Alon-Tarsi number of a graph. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001642The Prague dimension of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001780The order of promotion on the set of standard tableaux of given shape. St001829The common independence number of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001883The mutual visibility number of a graph. 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. St001917The order of toric promotion on the set of labellings of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000008The major index of the composition. St000017The number of inversions of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000089The absolute variation of a composition. St000117The number of centered tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000169The cocharge of a standard tableau. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000344The number of strongly connected outdegree sequences of a graph. St000348The non-inversion sum of a binary word. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000386The number of factors DDU in a Dyck path. St000466The Gutman (or modified Schultz) index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000660The number of rises of length at least 3 of a Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000691The number of changes of a binary word. St000741The Colin de Verdière graph invariant. St000768The number of peaks in an integer composition. St000778The metric dimension of a graph. St000846The maximal number of elements covering an element of a poset. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001596The number of two-by-two squares inside a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001657The number of twos in an integer partition. St001668The number of points of the poset minus the width of the poset. St001673The degree of asymmetry of an integer composition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001691The number of kings 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001730The number of times the path corresponding to a binary word crosses the base line. St001777The number of weak descents in an integer composition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001799The number of proper separations of a graph. St001827The number of two-component spanning forests of a graph. St001910The height of the middle non-run of a Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001949The rigidity index of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St001964The interval resolution global dimension of a poset. St000010The length of 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. St000087The number of induced subgraphs. St000088The row sums of the character table of the symmetric group. St000147The largest part of an integer partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. 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. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000277The number of ribbon shaped standard tableaux. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000331The number of upper interactions of a Dyck path. St000345The number of refinements of a partition. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000549The number of odd partial sums of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000636The hull number of a graph. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000785The number of distinct colouring schemes of a graph. St000808The number of up steps 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. St000820The number of compositions obtained by rotating the composition. 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. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000920The logarithmic height of a Dyck path. St000926The clique-coclique number of a graph. 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. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). 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. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001518The number of graphs with the same ordinary spectrum as the given graph. St001571The Cartan determinant of the integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001592The maximal number of simple paths between any two different vertices of a graph. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. 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. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001679The number of subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001720The minimal length of a chain of small intervals in a lattice. St001732The number of peaks visible from the left. St001739The number of graphs with the same edge polytope as the given graph. St001746The coalition number of a graph. St001754The number of tolerances of a finite lattice. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001765The number of connected components of the friends and strangers graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001802The number of endomorphisms of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001820The size of the image of the pop stack sorting operator. St001828The Euler characteristic of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001884The number of borders of a binary word. 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. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. 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. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000759The smallest missing part in an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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 $eA$. 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. 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: St001458The rank of the adjacency matrix of a graph. St001530The depth of a Dyck path. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000002The number of occurrences of the pattern 123 in a permutation. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000228The size of a partition. St000234The number of global ascents of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000459The hook length of the base cell of a partition. St000463The number of admissible inversions of a permutation. St000475The number of parts equal to 1 in a partition. St000478Another weight of a partition according to Alladi. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000779The tier of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. 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. St001115The number of even descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. 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. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra 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 St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001377The major index minus the number of inversions of a permutation. St001381The fertility of a permutation. St001394The genus of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001469The holeyness of a permutation. St001485The modular major index of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001513The number of nested exceedences of a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001651The Frankl number of a lattice. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001705The number of occurrences of the pattern 2413 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. 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). St001911A descent variant minus the number of inversions. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001960The number of descents of a permutation minus one if its first entry is not one. St000001The number of reduced words for a permutation. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000035The number of left outer peaks of a permutation. St000038The product of the heights of the descending steps of a Dyck path. St000056The decomposition (or block) number of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000108The number of partitions contained in the given partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000155The number of exceedances (also excedences) of a permutation. St000201The number of leaf nodes in a binary tree. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000245The number of ascents of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000288The number of ones in a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000413The number of ordered trees with the same underlying unordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000532The total number of rook placements on a Ferrers board. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000657The smallest part of an integer composition. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000678The number of up steps after the last double rise of a Dyck path. St000679The pruning number of an ordered tree. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000732The number of double deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000789The number of crossing-similar perfect matchings of a perfect matching. St000809The reduced reflection length of the permutation. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000816The number of standard composition tableaux of the composition. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000874The position of the last double rise in a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St000889The number of alternating sign matrices with the same antidiagonal sums. St000909The number of maximal chains of maximal size in a poset. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. 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 $c_{n−1}$. St000982The length of the longest constant subword. St000984The number of boxes below precisely one peak. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001052The length of the exterior of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001096The size of the overlap set of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. 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. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001268The size of the largest ordinal summand in the poset. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001281The normalized isoperimetric number of a graph. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001399The distinguishing number of a poset. St001400The total number of Littlewood-Richardson tableaux of given shape. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. 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). St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001684The reduced word complexity of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001779The order of promotion on the set of linear extensions of a poset. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001809The index of the step at the first peak of maximal height in a Dyck path. St001928The number of non-overlapping descents in a permutation. St001933The largest multiplicity of a part in an integer partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000007The number of saliances of the permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000312The number of leaves in a graph. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000673The number of non-fixed points of a permutation. St000676The number of odd rises of a Dyck path. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000842The breadth of a permutation. St000843The decomposition number of a perfect matching. St000844The size of the largest block in the direct sum decomposition of a permutation. St000922The minimal number such that all substrings of this length are unique. St000925The number of topologically connected components of a set partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000991The number of right-to-left minima of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001486The number of corners of the ribbon associated with an integer composition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000451The length of the longest pattern of the form k 1 2. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000422The energy of a graph, if it is integral. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001128The exponens consonantiae of a partition. St000302The determinant of the distance matrix of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000467The hyper-Wiener index of a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001626The number of maximal proper sublattices of a lattice. St000219The number of occurrences of the pattern 231 in a permutation. St000353The number of inner valleys of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) 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. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000711The number of big exceedences of a permutation. St000747A variant of the major index of a set partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000837The number of ascents of distance 2 of a permutation. 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. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000941The number of characters of the symmetric group whose value on the partition is even. St000961The shifted major index of a permutation. St001388The number of non-attacking neighbors of a permutation. St001557The number of inversions of the second entry of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000100The number of linear extensions of a poset. St000402Half the size of the symmetry class of a permutation. St000525The number of posets with the same zeta polynomial. St000530The number of permutations with the same descent word as the given permutation. St000568The hook number of a binary tree. St000633The size of the automorphism group of a poset. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000910The number of maximal chains of minimal length in a poset. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001246The maximal difference between two consecutive entries of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000893The number of distinct diagonal sums of an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000699The toughness times the least common multiple of 1,. 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. 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. St000403The Szeged index minus the Wiener index 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. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. 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. St001248Sum of the even parts of a partition. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-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. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001522The total irregularity of a graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001708The number of pairs of vertices of different degree in a graph. St001793The difference between the clique number and the chromatic number of a graph. 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. St000120The number of left tunnels of a Dyck path. St000145The Dyson rank of a partition. St000146The Andrews-Garvan crank of a partition. St000296The length of the symmetric border of a binary word. St000311The number of vertices of odd degree in a graph. St000327The number of cover relations in a poset. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000465The first Zagreb index of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000571The F-index (or forgotten topological index) of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000629The defect of a binary word. St000637The length of the longest cycle in a graph. St000658The number of rises of length 2 of a Dyck path. St000680The Grundy value for Hackendot on posets. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000807The sum of the heights of the valleys of the associated bargraph. St000915The Ore degree of a graph. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000995The largest even part of 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001371The length of the longest Yamanouchi prefix of a binary word. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001703The villainy of a graph. St001721The degree of a binary word. St001846The number of elements which do not have a complement in the lattice. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001930The weak major index of a binary word. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000297The number of leading ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. 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. St000693The modular (standard) major index of a standard tableau. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001955The number of natural descents for set-valued two row standard Young tableaux. St000045The number of linear extensions of a binary tree. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000782The indicator function of whether a given perfect matching is an L & P matching.