St001869The maximum cut size of a graph. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000189The number of elements in the poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000309The number of vertices with even degree. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000636The hull number of a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001029The size of the core of a graph. St001116The game chromatic 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. 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. St001330The hat guessing number 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. St001441The number of non-empty connected induced subgraphs of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000778The metric dimension of a graph. St001176The size of a partition minus its first part. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001345The Hamming dimension of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001458The rank of the adjacency matrix of a graph. St001644The dimension of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000063The number of linear extensions of a certain poset defined for an integer partition. St000081The number of edges of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000108The number of partitions contained in the given partition. St000228The size of a partition. St000271The chromatic index of a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000346The number of coarsenings of a partition. St000384The maximal part of the shifted composition of an integer partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000668The least common multiple of the parts of the partition. St000676The number of odd rises of a Dyck path. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. 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. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000867The sum of the hook lengths in the first row of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000912The number of maximal antichains in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000933The number of multipartitions of sizes given by an integer partition. St000947The major index east count of a Dyck path. St000983The length of the longest alternating subword. St001093The detour number of a graph. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001112The 3-weak dynamic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001249Sum of the odd parts of a partition. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001312Number of parabolic noncrossing partitions indexed by the composition. St001316The domatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001486The number of corners of the ribbon associated with an integer composition. St001527The cyclic permutation representation number of an integer partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001645The pebbling number of a connected graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. 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. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001827The number of two-component spanning forests 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. St000008The major index of the composition. St000080The rank of the poset. St000086The number of subgraphs. St000089The absolute variation of a composition. St000120The number of left tunnels of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000377The dinv defect of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000468The Hosoya index of a graph. St000518The number of distinct subsequences in a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000766The number of inversions of an integer composition. St000915The Ore degree of a graph. St000940The number of characters of the symmetric group whose value on the partition is zero. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001119The length of a shortest maximal path in a graph. St001175The size of a partition minus the hook length of the base cell. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001268The size of the largest ordinal summand in the poset. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001459The number of zero columns in the nullspace of a graph. 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. St001512The minimum rank 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. 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. St001777The number of weak descents in an integer composition. St001782The order of rowmotion on the set of order ideals of a poset. St001956The comajor index for set-valued two-row standard Young tableaux. St000656The number of cuts of a poset. St000004The major index of a permutation. St000007The number of saliances of the permutation. St000012The area of a Dyck path. St000013The height of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. 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. St000048The multinomial of the parts of a partition. St000051The size of the left subtree of a binary tree. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000144The pyramid weight of the Dyck path. St000153The number of adjacent cycles of a permutation. St000179The product of the hook lengths of the integer partition. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000315The number of isolated vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000519The largest length of a factor maximising the subword complexity. St000544The cop number of a graph. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. 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. St000794The mak of a permutation. St000797The stat`` of a permutation. St000808The number of up steps of the associated bargraph. 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. St000833The comajor index of a permutation. St000839The largest opener of a set partition. St000840The number of closers smaller than the largest opener in a perfect matching. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000984The number of boxes below precisely one peak. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000988The orbit size of a permutation under Foata's bijection. St000991The number of right-to-left minima 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. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of 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. St001034The area of the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001239The largest vector space dimension of the double dual of a simple module in 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001313The number of Dyck paths above the lattice path given by a binary word. St001315The dissociation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001363The Euler characteristic of a graph according to Knill. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. 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. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001759The Rajchgot index of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000074The number of special entries. St000090The variation of a composition. St000091The descent variation of a composition. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000133The "bounce" of a permutation. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000156The Denert index of a permutation. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000203The number of external nodes of a binary tree. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000290The major index of a binary word. St000295The length of the border of a binary word. St000305The inverse major index of a permutation. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000339The maf index of a permutation. St000359The number of occurrences of the pattern 23-1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000546The number of global descents of a permutation. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000682The Grundy value of Welter's game on a binary word. 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000732The number of double deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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 $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001023Number of simple modules with projective dimension at most 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. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). 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. St001090The number of pop-stack-sorts needed to sort a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001127The sum of the squares of the parts of a partition. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001251The number of parts of a partition that are not congruent 1 modulo 3. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001280The number of parts of an integer partition that are at least two. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001375The pancake length of a permutation. 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. St001438The number of missing boxes of a skew partition. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. 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. St001584The area statistic between a Dyck path and its bounce path. St001638The book thickness of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001671Haglund's hag of a permutation. St001684The reduced word complexity of a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. 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. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001706The number of closed sets in a graph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001592The maximal number of simple paths between any two different vertices of a graph. St000928The sum of the coefficients of the character polynomial of an integer partition. 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. St000061The number of nodes on the left branch of a binary tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000675The number of centered multitunnels of a Dyck path. St000729The minimal arc length of a set partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000823The number of unsplittable factors of the set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000990The first ascent of a permutation. St001062The maximal size of a block of a set partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000259The diameter of a connected graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step 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. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000796The stat' of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000467The hyper-Wiener index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000667The greatest common divisor of the parts of the partition. St000914The sum of the values of the Möbius function of a poset. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000015The number of peaks of a Dyck path. St000088The row sums of the character table of the symmetric group. 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. St000456The monochromatic index of a connected graph. St000526The number of posets with combinatorially isomorphic order polytopes. St000567The sum of the products of all pairs of parts. St000680The Grundy value for Hackendot on posets. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000717The number of ordinal summands of a poset. St000759The smallest missing part in an integer partition. St000762The sum of the positions of the weak records of an integer composition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. 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. St000904The maximal number of repetitions of an integer composition. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000992The alternating sum of the parts of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001060The distinguishing index of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama 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: St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001675The number of parts equal to the part in the reversed composition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001881The number of factors of a lattice as a Cartesian product of lattices. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000422The energy of a graph, if it is integral. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000014The number of parking functions supported by a Dyck path. St000117The number of centered tunnels of a Dyck path. St000391The sum of the positions of the ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000477The weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000674The number of hills of a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000753The Grundy value for the game of Kayles on a binary word. St000770The major index of an integer partition when read from bottom to top. St000792The Grundy value for the game of ruler on a binary word. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000976The sum of the positions of double up-steps of a Dyck path. St000981The length of the longest zigzag subpath. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001564The value of the forgotten symmetric functions when all variables set to 1. St001808The box weight or horizontal decoration of a Dyck path. St001885The number of binary words with the same proper border set. St001910The height of the middle non-run of a Dyck path. St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001378The product of the cohook lengths of the integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000455The second largest eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St000038The product of the heights of the descending steps of a Dyck path. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000289The decimal representation of a binary word. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000347The inversion sum of a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000627The exponent of a binary word. St000644The number of graphs with given frequency partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000869The sum of the hook lengths of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001658The total number of rook placements on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001915The size of the component corresponding to a necklace in Bulgarian solitaire.