Your data matches 908 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000228: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
St000459: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
St000460: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
St000870: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
St001380: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
Description
The number of monomer-dimer tilings of a Ferrers diagram. For a hook of length $n$, this is the $n$-th Fibonacci number.
St001382: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 1 = 2 - 1
[3]
=> 2 = 3 - 1
[2,1]
=> 2 = 3 - 1
[1,1,1]
=> 2 = 3 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 1
[2]
=> 100 => 2
[1,1]
=> 110 => 2
[3]
=> 1000 => 3
[2,1]
=> 1010 => 3
[1,1,1]
=> 1110 => 3
Description
The number of inversions of a binary word.
Mp00095: Integer partitions —to binary word⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 1
[2]
=> 100 => 2
[1,1]
=> 110 => 2
[3]
=> 1000 => 3
[2,1]
=> 1010 => 3
[1,1,1]
=> 1110 => 3
Description
The largest length of a factor maximising the subword complexity. Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal: $$ H_w = \max\{n: p_w(n)\text{ is maximal}\} $$ A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Mp00095: Integer partitions —to binary word⟶ Binary words
St000922: Binary words ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 1
[2]
=> 100 => 2
[1,1]
=> 110 => 2
[3]
=> 1000 => 3
[2,1]
=> 1010 => 3
[1,1,1]
=> 1110 => 3
Description
The minimal number such that all substrings of this length are unique.
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> 3
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
The following 898 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001034The area of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St000921The number of internal inversions of a binary word. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001437The flex of a binary word. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001065Number of indecomposable reflexive modules 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. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000050The depth or height of a binary tree. St000144The pyramid weight of the Dyck path. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000189The number of elements in the poset. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000246The number of non-inversions of a permutation. St000288The number of ones in a binary word. St000290The major index of a binary word. St000336The leg major index of a standard tableau. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000458The number of permutations obtained by switching adjacencies or successions. St000494The number of inversions of distance at most 3 of a permutation. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000617The number of global maxima of a Dyck path. St000657The smallest part of an integer composition. St000676The number of odd rises of a Dyck path. 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. St000719The number of alignments in a perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000784The maximum of the length and the largest part of the integer partition. St000808The number of up steps of the associated bargraph. St000809The reduced reflection length of the permutation. St000863The length of the first row of the shifted shape of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000924The number of topologically connected components of a perfect matching. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000957The number of Bruhat lower covers of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001018Sum of projective dimension of the 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. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. 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. 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. 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 $L=[c_0,c_1,...,c_{nāˆ’1}]$ 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: 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. 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. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001759The Rajchgot index of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000026The position of the first return of a Dyck path. St000044The number of vertices of the unicellular map given by a perfect matching. St000053The number of valleys of the Dyck path. St000058The order of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000070The number of antichains in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000108The number of partitions contained in the given partition. St000151The number of facets in the chain polytope of the poset. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000673The number of non-fixed points of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000844The size of the largest block in the direct sum decomposition of a permutation. 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. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nāˆ’1}]$ by adding $c_0$ to $c_{nāˆ’1}$. 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. St001012Number of simple modules with projective dimension at most 2 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001160The number of proper blocks (or intervals) of a permutations. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global 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 $eA$. 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. 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. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001245The cyclic maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001274The number of indecomposable injective modules with projective dimension equal to two. 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. 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001955The number of natural descents for set-valued two row standard Young tableaux. St000806The semiperimeter of the associated bargraph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001468The smallest fixpoint of a permutation. St000004The major index of a permutation. St000008The major index of the composition. St000013The height of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000025The number of initial rises 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. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000057The Shynar inversion number of a standard tableau. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000067The inversion number of the alternating sign matrix. St000075The orbit size of a standard tableau under promotion. St000081The number of edges of a graph. St000086The number of subgraphs. St000087The number of induced subgraphs. St000088The row sums of the character table of the symmetric group. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000117The number of centered tunnels of a Dyck path. St000134The size of the orbit of an alternating sign matrix under gyration. St000135The number of lucky cars of the parking function. St000136The dinv of a parking function. St000141The maximum drop size of a permutation. St000145The Dyson rank of a partition. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000197The number of entries equal to positive one in the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000203The number of external nodes of a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000224The sorting index of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000251The number of nonsingleton blocks of a set partition. St000258The burning number of a graph. St000271The chromatic index of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000295The length of the border of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000305The inverse major index of a permutation. St000308The height of the tree associated to a permutation. St000309The number of vertices with even degree. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000327The number of cover relations in a poset. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000339The maf index of a permutation. St000345The number of refinements of a partition. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000391The sum of the positions of the ones in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. 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. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000446The disorder of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000493The los statistic of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000504The cardinality of the first block of a set partition. St000507The number of ascents of a standard tableau. St000522The number of 1-protected nodes of a rooted tree. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000544The cop number of a graph. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000567The sum of the products of all pairs of parts. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000636The hull number of a graph. St000653The last descent of a permutation. St000654The first descent of a permutation. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000691The number of changes of a binary word. St000696The number of cycles in the breakpoint graph of a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000728The dimension of a set partition. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000753The Grundy value for the game of Kayles on a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000780The size of the orbit under rotation of a perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000794The mak of a permutation. St000795The mad of a permutation. St000797The stat`` of a permutation. St000798The makl of a 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. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000847The number of standard Young tableaux whose descent set is the binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. 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. 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. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000945The number of matchings in the dihedral orbit of a perfect matching. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000956The maximal displacement of a permutation. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000991The number of right-to-left minima of a permutation. St000992The alternating sum of the parts of an integer partition. 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. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001008Number of indecomposable injective 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001033The normalized 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. St001052The length of the exterior of a permutation. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001062The maximal size of a block of a set partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001075The minimal size of a block of a set partition. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). 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. St001093The detour number of a graph. St001094The depth index of a set partition. St001096The size of the overlap set of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. 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. 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 $g$. 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. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001246The maximal difference between two consecutive entries of a permutation. St001268The size of the largest ordinal summand in the poset. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001286The annihilation 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. St001308The number of induced paths on three vertices in a graph. 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. St001322The size of a minimal independent dominating set in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001345The Hamming dimension 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. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. 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. St001397Number of pairs of incomparable elements in a finite poset. St001401The number of distinct entries in a semistandard tableau. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001428The number of B-inversions of a signed permutation. St001430The number of positive entries in a signed permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001441The number of non-empty connected induced subgraphs of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001479The number of bridges of a graph. St001481The minimal height of a peak of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. 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. 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). St001517The length of a longest pair of twins in a permutation. St001521Half the total irregularity of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001527The cyclic permutation representation number of an integer partition. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001541The Gini index of an integer partition. St001566The length of the longest arithmetic progression in a permutation. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. 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. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001671Haglund's hag of a permutation. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a 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. St001725The harmonious chromatic number of a graph. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed 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. St001801Half the number of preimage-image pairs of different parity in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001827The number of two-component spanning forests of a graph. 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. St001861The number of Bruhat lower covers of a permutation. St001869The maximum cut size of a graph. St001874Lusztig's a-function for the symmetric group. St001910The height of the middle non-run of a Dyck path. St001917The order of toric promotion on the set of labellings of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001925The minimal number of zeros in a row of an alternating sign matrix. St001959The product of the heights of the peaks of a Dyck path. St000007The number of saliances of the permutation. St000012The area of a Dyck path. St000017The number of inversions of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000047The number of standard immaculate tableaux of a given shape. St000054The first entry of the permutation. St000094The depth of an ordered tree. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000167The number of leaves of an ordered tree. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000215The number of adjacencies of a permutation, zero appended. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000231Sum of the maximal elements of the blocks of a set partition. St000234The number of global ascents of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000245The number of ascents of a permutation. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000259The diameter of a connected graph. St000291The number of descents of a binary word. St000294The number of distinct factors of a binary word. St000325The width of the tree associated to 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. St000358The number of occurrences of the pattern 31-2. 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. St000377The dinv defect of an integer partition. St000389The number of runs of ones of odd length in a binary word. 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. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000470The number of runs in a permutation. St000472The sum of the ascent bottoms of a permutation. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000490The intertwining number of a set partition. St000505The biggest entry in the block containing the 1. St000518The number of distinct subsequences in a binary word. St000521The number of distinct subtrees of an ordered tree. St000538The number of even inversions of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000624The normalized sum of the minimal distances to a greater element. St000625The sum of the minimal distances to a greater element. St000651The maximal size of a rise in a permutation. St000656The number of cuts of a poset. St000662The staircase size of the code of a permutation. St000668The least common multiple of the parts of the partition. 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. St000692Babson and SteingrĆ­msson's statistic of a permutation. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000741The Colin de VerdiĆØre graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000770The major index of an integer partition when read from bottom to top. St000778The metric dimension of a graph. St000779The tier of a permutation. St000820The number of compositions obtained by rotating the composition. St000825The sum of the major and the inverse major index of a permutation. St000836The number of descents of distance 2 of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000881The number of short braid edges in the graph of braid moves of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000891The number of distinct diagonal sums of a permutation matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000915The Ore degree of a graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000946The sum of the skew hook positions in a Dyck path. St000971The smallest closer of a set partition. St000973The length of the boundary of an ordered tree. St000984The number of boxes below precisely one peak. St000989The number of final rises 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. 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. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001259The vector space dimension of the double dual of D(A) in 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. St001279The sum of the parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001379The number of inversions plus the major index of a permutation. St001402The number of separators in a permutation. St001405The number of bonds in a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in 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. St001469The holeyness of a permutation. 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. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001623The number of doubly irreducible elements of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001649The length of a longest trail in a graph. St001658The total number of rook placements on a Ferrers board. St001684The reduced word complexity 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. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001814The number of partitions interlacing the given partition. St001826The maximal number of leaves on a vertex of a graph. St001834The number of non-isomorphic minors of a graph. St001927Sparre Andersen's number of positives of a signed permutation. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001949The rigidity index of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000400The path length of an ordered tree. St000438The position of the last up step in a Dyck path. St000520The number of patterns in a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000928The sum of the coefficients of the character polynomial of an integer partition. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000301The number of facets of the stable set polytope of a graph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. 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 $eA$. St000071The number of maximal chains in a poset. St000402Half the size of the symmetry class of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000675The number of centered multitunnels of a Dyck path. St000909The number of maximal chains of maximal size in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000941The number of characters of the symmetric group whose value on the partition is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001388The number of non-attacking neighbors of a permutation. St001930The weak major index of a binary word. St000064The number of one-box pattern of a permutation. St000461The rix statistic of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000815The number of semistandard Young tableaux of partition weight of given shape. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000933The number of multipartitions of sizes given by an integer partition. St001060The distinguishing index of a graph. 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. St001409The maximal entry of a semistandard tableau. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001926Sparre Andersen's position of the maximum of a signed permutation. St000020The rank of the permutation. St000040The number of regions of the inversion arrangement of a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000241The number of cyclical small excedances. St000296The length of the symmetric border of a binary word. St000401The size of the symmetry class of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000478Another weight of a partition according to Alladi. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000638The number of up-down runs of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000874The position of the last double rise in a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001128The exponens consonantiae of a partition. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001482The product of the prefix sums of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001884The number of borders of a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001948The number of augmented double ascents of a permutation. St000055The inversion sum of a permutation. St000137The Grundy value of an integer partition. St000238The number of indices that are not small weak excedances. St000347The inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000628The balance of a binary word. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000710The number of big deficiencies of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. 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. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001375The pancake length of a permutation. St001383The BG-rank of an integer partition. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. 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. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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 $eA$. St001535The number of cyclic alignments of a permutation. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000699The toughness times the least common multiple of 1,. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001565The number of arithmetic progressions of length 2 in a permutation. St001626The number of maximal proper sublattices of a lattice. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001651The Frankl number of a lattice. 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. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000910The number of maximal chains of minimal length in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001902The number of potential covers of a poset. St000045The number of linear extensions of a binary tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000739The first entry in the last row of a semistandard tableau. St001569The maximal modular displacement of a permutation. St001645The pebbling number of a connected graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of a permutation. St000842The breadth of a permutation. St000990The first ascent of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001516The number of cyclic bonds of a permutation. St001556The number of inversions of the third entry of a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000677The standardized bi-alternating inversion number of a permutation. St000690The size of the conjugacy class of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000796The stat' of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St001061The number of indices that are both descents and recoils of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001114The number of odd descents of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001346The number of parking functions that give the same permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000226The convexity of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000353The number of inner valleys of 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. St000471The sum of the ascent tops of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 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. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000934The 2-degree of an integer partition. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001731The factorization defect of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000652The maximal difference between successive positions of a permutation.