St000183The side length of the Durfee square of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St000146The Andrews-Garvan crank of a partition. St000474Dyson's crank of a partition. St000013The height of a Dyck path. St000053The number of valleys of the Dyck path. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000331The number of upper interactions of a Dyck path. St000378The diagonal inversion number of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in 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. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001884The number of borders of a binary word. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000142The number of even parts of a partition. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000225Difference between largest and smallest parts in a partition. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000519The largest length of a factor maximising the subword complexity. St000549The number of odd partial sums of an integer partition. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000759The smallest missing part in an integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. 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. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001214The aft of an integer partition. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001530The depth of a Dyck path. St001657The number of twos in an integer 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 $c_{n-1}$. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000086The number of subgraphs. St000172The Grundy number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. 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. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. 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. St000822The Hadwiger number of the graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001093The detour number of a graph. 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. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity 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. St001330The hat guessing number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001368The number of vertices of maximal degree in 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. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001618The cardinality of the Frattini sublattice of a lattice. St001624The breadth of a lattice. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000081The number of edges of a graph. St000171The degree of the graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. 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. St000310The minimal degree of a vertex of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000846The maximal number of elements covering an element of 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. St001071The beta invariant of the graph. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001175The size of a partition minus the hook length of the base cell. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a 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. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001525The number of symmetric hooks on the diagonal of a partition. 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. St001621The number of atoms of a lattice. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001689The number of celebrities in 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. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001962The proper pathwidth of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000143The largest repeated part of a partition. 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. St000259The diameter of a connected graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000364The exponent of the automorphism group of a graph. St000443The number of long tunnels of a Dyck path. St000456The monochromatic index of a connected graph. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000636The hull number of a graph. St000644The number of graphs with given frequency partition. St000668The least common multiple of the parts of the partition. 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. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000948The chromatic discriminant of a graph. 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. St001057The Grundy value of the game of creating an independent set in a graph. St001119The length of a shortest maximal path in a graph. St001128The exponens consonantiae of a partition. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001592The maximal number of simple paths between any two different vertices of a graph. St001642The Prague dimension of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000008The major index of the composition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000260The radius of a connected graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. 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. 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 $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001521Half the total irregularity of a graph. St001587Half of the largest even part of an integer partition. St001613The binary logarithm of the size of the center of a lattice. 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. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001736The total number of cycles in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001783The number of odd automorphisms of a graph. St001799The number of proper separations of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001827The number of two-component spanning forests of a graph. 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). St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000351The determinant of the adjacency matrix of a graph. St000477The weight of a partition according to Alladi. St000633The size of the automorphism group of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000640The rank of the largest boolean interval in a poset. 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. St000914The sum of the values of the Möbius function of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001118The acyclic chromatic index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001720The minimal length of a chain of small intervals in a lattice. St001959The product of the heights of the peaks of a Dyck path. St000095The number of triangles of a graph. St000100The number of linear extensions of a poset. St000264The girth of a graph, which is not a tree. St000286The number of connected components of the complement of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000635The number of strictly order preserving maps of a poset into itself. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. 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 $eA$. 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. St001480The number of simple summands of the module J^2/J^3. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001890The maximum magnitude of the Möbius function of a poset. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.