Statistics database

**Alternating sign matrices**(12 statistics) # matrix like objects

The number of entries equal to $-1$ in the alternating sign matrix.

The column of the unique '1' in the first row of the alternating sign matrix.

The inversion number of the alternating sign matrix.

The rank of the alternating sign matrix in the alternating sign matrix poset.

The size of the orbit of an alternating sign matrix under gyration.

The determinant of an alternating sign matrix.

The row of the unique '1' in the first column of the alternating sign matrix.

The number of entries equal to positive one in the alternating sign matrix.

The column of the unique '1' in the last row of the alternating sign matrix.

The row of the unique '1' in the last column of the alternating sign matrix.

The osculating paths major index of an alternating sign matrix.

The positive inversions of an alternating sign matrix.

**Binary trees**(35 statistics) # tree like structures # Catalan objects # graph like objects

The number of linear extensions of a binary tree.

The depth/height of a binary tree.

The size of the left subtree of a binary tree.

The number of nodes on the left branch of a binary tree.

The number of elements smaller than a binary tree in Tamari order.

The number of left oriented leafs of a binary tree except the first one.

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....

The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....

The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....

The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....

The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....

The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....

The sum of the sizes of the right subtrees of a binary tree.

The number of occurrences of the contiguous pattern [[.,.],[.,.

A decimal representation of a binary tree as a code word.

The number of leaf nodes in a binary tree.

The number of external nodes of a binary tree.

The number of internal nodes of a binary tree.

The number of nodes of degree 3 of a binary tree.

The number of vertices with out-degree 1 in a binary tree.

The register function (or Horton-Strahler number) of a binary tree.

The sum of the depths of the vertices (or total internal path length) of a binary....

The external path length of a binary tree.

The number of pitchforks in a binary tree.

The tree factorial of a binary tree.

The number of binary trees with the same underlying unordered tree.

The binary logarithm of the number of binary trees with the same underlying unord....

The hook number of a binary tree.

The sum of the heights of the vertices of a binary tree.

The protection number of a binary tree.

**Binary words**(35 statistics) # word like objects # path like objects

The number of ones in a binary word.

The decimal representation of a binary word.

The major index of a binary word.

The number of descents of a binary word.

The number of ascents of a binary word.

The number of inversions of a binary word.

The number of distinct factors of a binary word.

The length of the border of a binary word.

The length of the symmetric border of a binary word.

The number of leading ones in a binary word.

The position of the first one in a non-zero binary word.

The inversion sum of a binary word.

The non-inversion sum of a binary word.

The number of runs of ones of odd length in a binary word.

The number of runs of ones in a binary word.

The sum of the positions of the ones in a binary word.

The length of the longest run of ones in a binary word.

The number of strictly increasing runs in a binary word.

The number of distinct subsequences in a binary word.

The largest length of a factor maximising the subword complexity.

The number of permutations whose descent word is the given binary word.

The size of the conjugacy class of a binary word.

The minimal period of a binary word.

The exponent of a binary word.

The balance of a binary word.

The defect of a binary word.

The length of the shortest palindromic decomposition of a binary word.

The number of distinct palindromic decompositions of a binary word.

The Grundy value of Welter's game on a binary word.

The number of changes of a binary word.

The Grundy value for the game of Kayles on a binary word.

The Grundy value for the game of ruler on a binary word.

The stopping time of the decimal representation of the binary word for the 3x+1 p....

The decimal representation of a binary word with a leading 1.

The number of standard Young tableaux whose descent set is the binary word.

**Cores**(5 statistics) # partition like objects

**Dyck paths**(60 statistics) # Catalan objects # path like objects

The bounce statistic of a Dyck path.

The dinv statistic of a Dyck path.

The number of touch points of a Dyck path.

The area of a Dyck path.

The height of a Dyck path.

The number of parking functions supported by a Dyck path.

The number of peaks of a Dyck path.

The number of double rises of a Dyck path.

The number of initial rises of a Dyck path.

The position of the first return of a Dyck path.

The major index of a Dyck path.

The number of elements smaller than the given Dyck path in the Tamari Order.

The product of the heights of the descending steps of a Dyck path.

The number of valleys of a Dyck path not on the x-axis.

The number of valleys of the Dyck path.

The number of alternating sign matrices for a given Dyck path.

The number of centered tunnels of a Dyck path.

The number of left tunnels of a Dyck path.

The pyramid weight of the Dyck path.

The bounce count of a Dyck path.

The number of evenly positioned ascents, with the initial position equal to 1.

The number of upper interactions of a Dyck path.

The difference of lower and upper interactions.

The number of non-final maximal sub-paths of length greater than one.

The dinv deficit of a Dyck path.

The bounce deficit of a Dyck path.

The number of factors DDU in a Dyck path.

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

The sum of the heights of the peaks of a Dyck path.

The number of Dyck paths that are weakly below a Dyck path.

The number of Dyck paths that are weakly above a Dyck path, except for the path i....

The number of Dyck paths that are weakly above a Dyck path.

The number of Dyck paths that are weakly below a Dyck path, except for the path i....

The position of the last up step in a Dyck path.

The position of the first down step of a Dyck path.

The maximal area to the right of an up step of a Dyck path.

The number of long tunnels of a Dyck path.

The length of the maximal rise of a Dyck path.

The number of rises of length 1 of a Dyck path.

The sum of the semi-lengths of tunnels before a valley of a Dyck path.

The number of global maxima of a Dyck path.

The sum of the areas of the rectangles formed by two consecutive peaks and the va....

The length of the minimal rise of a Dyck path.

The number of rises of length 2 of a Dyck path.

The number of rises of length at least 2 of a Dyck path.

The number of rises of length at least 3 of a Dyck path.

The number of rises of length 3 of a Dyck path.

The number of hills of a Dyck path.

The number of centered multitunnels of a Dyck path.

The number of odd rises of a Dyck path.

The number of up steps after the last double rise of a Dyck path.

The number of points below the Dyck path such that the diagonal to the north-east....

The global dimension of the LNakayama algebra associated to a Dyck path.

The dominant dimension of the LNakayama algebra associated to a Dyck path.

The finitistic dominant dimension of a Dyck path.

The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

The global dimension minus the dominant dimension of the LNakayama algebra associ....

The maximal n such that the minimal generator-cogenerator module in the LNakayama....

The number of pairs of centered tunnels, one strictly containing the other, of a ....

The number of pairs of left tunnels, one strictly containing the other, of a Dyck....

**Finite Cartan types**(7 statistics) # Cartan type objects

The size of the associated Weyl group.

The dimension of the representation $V(\Lambda_1)$.

The rank of the Cartan type.

The Catalan number of an irreducible finite Cartan type.

The Coxeter number of a finite Cartan type.

The positive Catalan number of an irreducible finite Cartan type.

The determinant of the Cartan matrix.

**Gelfand-Tsetlin patterns**(11 statistics) # matrix like objects

The number of circled entries.

The number of boxed entries.

The number of special entries.

The number of boxed and circled entries.

The sum of the entries of the Gelfand-Tsetlin pattern.

The single entry in the last row.

The number of boxed plus the number of special entries.

The total number of tiles in the Gelfand-Tsetlin pattern.

The number of free tiles in the pattern.

Number of free entries.

The sum of the first row in a Gelfand-Tsetlin pattern.

**Graphs**(104 statistics) # graph like objects

The number of edges of a graph.

The number of subgraphs.

The number of induced subgraphs.

The length of the maximal independent set of vertices of a graph.

The number of triangles of a graph.

The number of spanning trees of a graph.

The order of the largest clique of the graph.

The chromatic number of a graph.

The degree of the graph.

The Grundy number of a graph.

The cardinality of the automorphism group of a graph.

The burning number of a graph.

The diameter of a connected graph.

The radius of a connected graph.

The edge connectivity of a graph.

The vertex connectivity of a graph.

The Szeged index of a graph.

The girth of a graph, which is not a tree.

The Wiener index of a graph.

The number of spanning subgraphs of a graph with the same connected components.

The number of maximal spanning forests contained in a graph.

The number of strongly connected orientations of a graph.

The number of acyclic orientations of a graph.

The number of forests contained in a graph.

The chromatic index of a connected graph.

The treewidth of a graph.

The domination number of a graph.

The number of perfect matchings of a graph.

The size of the preimage of the map 'to graph' from Ordered trees to Graphs.

The size of the preimage of the map 'to graph' from Binary trees to Graphs.

The number of connected components of the complement of a graph.

The number of connected components of a graph.

The number of nonisomorphic vertex-induced subtrees.

The number of independent sets of vertices of a graph.

The number of facets of the stable set polytope of a graph.

The determinant of the distance matrix of a connected graph.

The determinant of the product of the incidence matrix and its transpose of a gra....

The number of vertices with even degree.

The minimal degree of a vertex of a graph.

The number of vertices of odd degree in a graph.

The number of leaves in a graph.

The number of degree 2 vertices of a graph.

The number of isolated vertices of a graph.

The skewness of a graph.

The minimal crossing number of a graph.

The number of spanning subgraphs of a graph.

The number of strongly connected outdegree sequences of a graph.

The number of different adjacency matrices of a graph.

The sum of the vertex degrees of a graph.

The determinant of the adjacency matrix of a graph.

The second Zagreb index of a graph.

The size of a minimal vertex cover of a graph.

The number of minimal vertex covers of a graph.

The exponent of the automorphism group of a graph.

The Altshuler-Steinberg determinant of a graph.

The genus of a graph.

The number of Hamiltonian cycles in a graph.

The matching number of a graph.

The number of orbits of vertices of a graph under automorphisms.

The Szeged index minus the Wiener index of a graph.

The energy of a graph, if it is integral.

The number of pairs of vertices of a graph with distance 3.

The number of pairs of vertices of a graph with distance 2.

The number of pairs of vertices of a graph with distance 4.

The number of edges minus the number of vertices plus 2 of a graph.

The number of distinct eigenvalues of a graph.

The number of distinct Laplacian eigenvalues of a graph.

The largest eigenvalue of a graph if it is integral.

The second largest eigenvalue of a graph if it is integral.

The monochromatic index of a connected graph.

The Schultz index of a connected graph.

The first Zagreb index of a graph.

The Gutman (or modified Schultz) index of a connected graph.

The hyper-Wiener index of a connected graph.

The Hosoya index of a graph.

The distinguishing number of a graph.

The Ramsey number of a graph.

The (zero)-forcing number of a graph.

The rank-width of a graph.

The pathwidth of a graph.

The cutwidth of a graph.

The cop number of a graph.

The number of cut vertices of a graph.

The number of blocks of a connected graph.

The F-index (or forgotten topological index) of a graph.

The hull number of a graph.

The length of the longest cycle in a graph.

The maximin edge-connectivity for choosing a subgraph.

The toughness times the least common multiple of 1,.

The largest Laplacian eigenvalue of a graph if it is integral.

The number of different neighbourhoods in a graph.

The maximal cardinality of a set of vertices with the same neighbourhood in a gra....

The Colin de VerdiÃ¨re graph invariant.

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.....

The multiplicity of the largest distance Laplacian eigenvalue in a connected grap....

The multiplicity of the largest Laplacian eigenvalue in a graph.

The maximal multiplicity of a Laplacian eigenvalue in a graph.

The multiplicity of the largest eigenvalue in a graph.

The maximal multiplicity of an eigenvalue in a graph.

The number of distinct eigenvalues of the distance Laplacian of a connected graph....

The metric dimension of a graph.

The number of distinct colouring schemes of a graph.

The maximal number of occurrences of a colour in a proper colouring of a graph.

The Hadwiger number of the graph.

**Integer compositions**(31 statistics) # partition like objects

The major index of the composition.

The number of standard immaculate tableaux of a given shape.

The absolute variation of a composition.

The variation of a composition.

The descent variation of a composition.

The size of the preimage of the map 'descent composition' from Permutations to In....

The size of the preimage of the map 'to inverse des composition' from Parking fun....

The largest part of an integer composition.

The first part of an integer composition.

The last part of an integer composition.

The smallest part of an integer composition.

The length of the longest weakly inreasing subsequence of parts of an integer com....

The length of the longest staircase fitting into an integer composition.

The length of the longest strictly decreasing subsequence of parts of an integer ....

The number of ascents in an integer composition.

The sum of the positions of the weak records of an integer composition.

The sum of the positions of the strong records of an integer composition.

The number of strong records in an integer composition.

The number of weak records in an integer composition.

The number of inversions of an integer composition.

The number of runs in an integer composition.

The number of peaks in an integer composition.

The major index of a composition.

The number of peaks of the associated bargraph.

The semiperimeter of the associated bargraph.

The sum of the heights of the valleys of the associated bargraph.

The number of up steps of the associated bargraph.

The number of standard composition tableaux of the composition.

The sum of the entries in the column specified by the composition of the change o....

The sum of the entries in the column specified by the composition of the change o....

The number of compositions obtained by rotating the composition.

**Integer partitions**(107 statistics) # partition like objects

The number of standard Young tableaux of the partition.

The length of the partition.

The largest eigenvalue of the random to random operator acting on the simple modu....

The multinomial of the parts of a partition.

The number of set partitions whose sorted block sizes correspond to the partition....

The number of linear extensions of a certain poset defined from a partition \la....

The row sums of the character table of the symmetric group.

The number of partitions contained in the given partition.

The Grundy value of an integer partition.

The number of even parts of a partition.

The maximal repeated part of a partition.

The Dyson rank of a partition.

The Andrews-Garvan crank of a partition.

The largest part of an integer partition.

The number of odd parts of a partition.

The number of cells of the partition whose leg is zero and arm is odd.

The floored half-sum of the multiplicities of a partition.

The number of different parts of an integer partition.

Multiplicity of the smallest part of $\lambda$.

Degree of the polynomial counting the number of semistandard Young-tableaux of sh....

Product of hook lengths.

The number of permutations whose cycle type is the given integer partition.

The size of the Durfee Square of a partition.

The size of the centralizer of any permutation of given cycle type.

The weighted size of a partition.

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

Number of standard Young tableaux for an integer partition such that no two conse....

Difference between largest and smallest parts in a partition.

The size of a partition.

The number of parts from which one can substract 2 and still get an integer parti....

The number of distinct parts of a partition that occur at least twice.

Number of permutations whose sorted list of non zero multiplicities of the Lehmer....

The size of the preimage of the map 'to partition' from Integer compositions to I....

The size of the preimage of the map 'Robinson-Schensted tableau shape' from Permu....

The number of addable cells of the Ferrers diagram of an integer partition.

The spin of an integer partition.

The dinv adjustment of an integer partition.

The number of integer partitions of n that are dominated by an integer partition.....

The number of refinements of a partition.

The number of coarsenings of a partition.

The dinv defect of an integer partition.

The diagonal inversion number of an integer partition.

Half the perimeter of the largest rectangle that fits inside the diagram of an in....

The maximal part of the shifted composition of an integer partition.

The hook length of the base cell of a partition.

The hook length of the last cell along the main diagonal of an integer partition.....

The number of parts of a partition that are strictly bigger than the number of on....

Dyson's crank of a partition.

The number of parts equal to 1 in a partition.

The weight of a partition according to Alladi.

Another weight of a partition according to Alladi.

The number of lower covers of a partition in dominance order.

The number of upper covers of a partition in dominance order.

The number of standard desarrangement tableaux of shape equal to the given partit....

The diagonal index (content) of a partition.

The number of invariant oriented cycles when acting with a permutation of given c....

The number of invariant subsets when acting with a permutation of given cycle typ....

The number of invariant subsets of size 3 when acting with a permutation of given....

The number of invariant subsets of size 2 when acting with a permutation of given....

The number of invariant simple graphs when acting with a permutation of given cyc....

The number of invariant set partitions when acting with a permutation of given cy....

The Kreweras number of an integer partition.

The number of ways to place as many non-attacking rooks as possible on a Ferrers ....

The total number of rook placements on a Ferrers board.

The maximal number of non-attacking rooks on a Ferrers shape.

The number of even non-empty partial sums of an integer partition.

The number of different non-empty partial sums of an integer partition.

The number of odd partial sums of an integer partition.

The number of ways to select a row of a Ferrers shape and two cells in this row.

The sum of the products of all pairs of parts.

The number of self-evacuating tableaux of given shape.

The number of standard tableaux of shape equal to the given partition such that t....

The number of standard tableaux of shape equal to the given partition such that t....

The number of graphs with given frequency partition.

The greatest common divisor of the parts of the partition.

The least common multiple of the parts of the partition.

The Grundy value of Chomp on Ferrers diagrams.

The number of 3-rim hooks removed from an integer partition to obtain its associa....

The number of 2-rim hooks removed from an integer partition to obtain its associa....

The number of semistandard tableaux on a given integer partition with minimal max....

The number of semistandard tableaux on a given integer partition of n with maxima....

The product of the factorials of the multiplicities of an integer partition.

The product of the factorials of the parts.

The product of the parts of an integer partition.

The number of semistandard Young tableau of given shape, with entries at most 4.

The dimension of the irreducible representation of Sp(4) labelled by an integer p....

The number of semistandard Young tableau of given shape, with entries at most 2.

The number of semistandard Young tableau of given shape, with entries at most 3.

The dimension of the irreducible representation of Sp(6) labelled by an integer p....

The smallest integer d such that the restriction of the representation correspond....

The Grundy value for the game 'Couples are forever' on an integer partition.

The number of real roots of the characteristic polynomial of a linear recurrence ....

The smallest missing part in an integer partition.

The major index of an integer partition when read from bottom to top.

The number of proper colouring schemes of a Ferrers diagram.

The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....

The maximum of the length and the number of parts of an integer partition.

The sum of the entries in the column specified by the partition of the change of ....

The sum of the entries in the column specified by the partition of the change of ....

The sum of the entries in the column specified by the partition of the change of ....

The number of zero-one matrices with weakly decreasing column sums and row sums g....

The sum of the entries in the column specified by the partition of the change of ....

The number of semistandard Young tableaux of partition weight of given shape.

The minimal difference in size when partitioning the integer partition into two s....

**Ordered trees**(19 statistics) # tree like structures # Catalan objects # graph like objects

The number of subtrees.

The number of linear extensions of the tree.

The depth of an ordered tree.

The depth minus 1 of an ordered tree.

The number of leaves of an ordered tree.

The number of internal nodes of an ordered tree.

The maximum number of child nodes in a tree.

The Strahler number of a rooted tree.

The path length of an ordered tree.

The tree factorial of an ordered tree.

The number of ordered trees with the same underlying unordered tree.

The size of the automorphism group of the rooted tree underlying the ordered tree....

The number of inequivalent increasing trees of an ordered tree.

The size of the automorphism group of the ordered tree.

The number of distinct subtrees of an ordered tree.

The number of 1-protected nodes of a rooted tree.

The number of 2-protected nodes of a rooted tree.

The pruning number of an ordered tree.

The protection number of an ordered tree.

**Parking functions**(7 statistics) # word like objects # path like objects

The number of lucky cars of the parking function.

The dinv of a parking function.

Sum of the entries.

The cosum of a parking function.

The number of primary dinversion pairs of a labelled dyck path corresponding to a....

The number of secondary dinversion pairs of the dyck path corresponding to a park....

The sum of the entries of a parking function minus its length.

**Perfect matchings**(20 statistics) # graph like objects

The number of nestings of a perfect matching.

The number of crossings of a perfect matching.

The number of crossings plus two-nestings of a perfect matching.

The number of vertices of the unicellular map given by a perfect matching.

The number of short pairs.

The number of alignments in a perfect matching.

The size of the largest partition in the oscillating tableau corresponding to the....

The sum of the partition sizes in the oscillating tableau corresponding to a perf....

The number of pairs with odd minimum in a perfect matching.

The Grundy value for the game of removing nestings in a perfect matching.

The size of the orbit under rotation of a perfect matching.

The indicator function of whether a given perfect matching is an L & P matching.

The number of flips required to make a perfect matching noncrossing.

The number of nesting-similar perfect matchings of a perfect matching.

The number of crossing-similar perfect matchings of a perfect matching.

The propagating number of a perfect matching.

The number of terminal right-hand endpoints when the vertices are written in orde....

The number of closers smaller than the largest opener in a perfect matching.

The largest opener of a perfect matching.

The decomposition number of a perfect matching.

**Permutations**(234 statistics) # word like objects

The number of ways to write a permutation as a minimal length product of simple t....

The number of occurrences of the pattern 123 in a permutation.

The major index of a permutation.

The number of saliances of the permutation.

The number of inversions of a permutation.

The cardinality of the complement of the connectivity set.

The rank of the permutation.

The number of descents of a permutation.

The number of fixed points of a permutation.

The number of inner peaks of a permutation.

The number of stack-sorts needed to sort a permutation.

The depth of a permutation.

The sum of the descent differences of a permutations.

The number of cycles in the cycle decomposition of a permutation.

The number of permutations greater than or equal to the given permutation in (str....

The maximum defect over any reduced expression for a permutation and any subexpre....

The number of left outer peaks of a permutation.

The evaluation of the Kazhdan-Lusztig polynomial $P(id,w)$ for each permutation $w$ i....

The sign of a permutation.

The number of crossings of a permutation.

The number of regions of inversion arrangement of a permutation.

The first entry of the permutation.

The inversion sum of a permutation.

The decomposition number of a permutation.

The order of a permutation.

The greater neighbor of the maximum.

The length of the longest increasing subsequence of the permutation.

The number of one-box pattern of a permutation.

The number of alternating sign matrices whose left key is the permutation.

The number of outer peaks of a permutation.

The number of valleys of a permutation, including the boundary.

The number of elements less than or equal to the given element in Bruhat order.

The number of permutations less than or equal to given permutation in left weak o....

The sum of the descent tops (or Genocchi descents) of a permutation.

The number of occurrences of the pattern 321 in a permutation.

The difference in Coxeter length of a permutation and its image under the Simion-....

The cardinality of the preimage of the Simion-Schmidt map.

The "bounce" of a permutation.

The maximum drop size of a permutation.

The number of adjacent cycles of a permutation.

The sum of the descent bottoms of a permutation.

The number of exceedances (also excedences) of a permutation.

The Denert index of a permutation.

The number of nontrivial cycles of a permutation $\pi$ in its cycle decomposition.

Maximum difference of elements in cycles.

Minimum over maximum difference of elements in cycles.

The number of weak exceedances (also weak excedences) of a permutation.

The number of adjacencies (or small descents) of a permutation.

The number of adjacencies of a permutation, 0 appended.

The absolute length of a permutation.

The number of occurrences of the pattern 312 in a permutation.

The number of occurrences of the pattern 213 in a permutation.

The number of occurrences of the pattern 231 in a permutation.

The number of occurrences of the pattern 132 in a permutation.

The number of strong fixed points of a permutation.

The number of alignments of a permutation

The number of nestings of a permutation

The sorting index of a permutation.

The convexity of a permutation.

The number of global ascents of a permutation.

The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.

The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

The number of indices $i$ such that $\pi_i=i+1$.

The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.

The number of indices $i$ for which $\pi_i \neq i+1$.

The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.

The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.....

The number of cyclic valleys and cyclic peaks of a permutation.

The number of ascents of a permutation.

The number of non-inversions of a permutation.

The number of reduced Kogan faces with the permutation as type.

The size of the preimage of the map 'cycle-as-one-line notation' from Permutation....

The size of the preimage of the map 'to labelling permutation' from Parking funct....

The load of a permutation.

The inverse major index of a permutation.

The height of the tree associated to a permutation.

The number of left-to-right-maxima of a permutation.

The number of non-left-to-right-maxima of a permutation.

The cycle descent number of a permutation.

The shape of the tree associated to a permutation.

The width of the tree associated to a permutation.

The dez statistic, the number of descents of a permutation after replacing fixed ....

The maz index, the major index of a permutation after replacing fixed points by z....

The lec statistic, the sum of the inversion numbers of the hook factors of a perm....

The number of pixed points of a permutation.

The maf index of a permutation.

The non-inversion sum of a permutation.

The cosine of a permutation.

The Elizalde-Pak rank of a permutation.

The number of inner valleys of a permutation.

The number of recoils of a permutation.

The number of occurrences of the pattern 21-3.

The number of occurrences of the pattern 13-2.

The number of occurrences of the pattern 12-3.

The number of occurrences of the pattern 31-2.

The number of occurrences of the pattern 23-1.

The number of occurrences of the pattern 32-1.

The number of double ascents of a permutation.

The number of double descents of a permutation.

The number of simsun double descents of a permutation.

The number of positions mid points of decreasing subsequences of length 3 in a pe....

The number of positions of mid points of increasing subsequences of length 3 in a....

The number of weak exceedences of a permutation that are also mid-points of a dec....

The number of non weak exceedences of a permutation that are also not mid-points ....

The number of non weak exceedences of a permutation that are mid-points of a decr....

The size of the symmetry class of a permutation.

Half the size of the symmetry class of a permutation.

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permuta....

The number of occurrences of the pattern 1324 in a permutation.

The number of occurrences of the pattern 3241 in a permutation.

The number of occurrences of the pattern 2143 in a permutation.

The number of occurrences of the pattern 4231 in a permutation.

The number of occurrences of the pattern 123 or of the pattern 132 in a permutati....

The number of occurrences of the pattern 132 or of the pattern 231 in a permutati....

The number of occurrences of the pattern 132 or of the pattern 213 in a permutati....

The number of occurrences of the pattern 132 or of the pattern 312 in a permutati....

The number of occurrences of the pattern 123 or of the pattern 231 in a permutati....

The number of occurrences of the pattern 123 or of the pattern 213 in a permutati....

The number of occurrences of the pattern 123 or of the pattern 321 in a permutati....

The number of occurrences of the pattern 123 or of the pattern 312 in a permutati....

The number of occurrences of the pattern 213 or of the pattern 321 in a permutati....

The number of occurrences of the pattern 231 or of the pattern 312 in a permutati....

The number of occurrences of the pattern 132 or of the pattern 321 in a permutati....

The number of occurrences of the pattern 213 or of the pattern 312 in a permutati....

The number of occurrences of the pattern 213 or of the pattern 231 in a permutati....

The number of occurrences of the pattern 231 or of the pattern 321 in a permutati....

The number of occurrences of the pattern 312 or of the pattern 321 in a permutati....

The number of occurrences of the pattern 4132 or of the pattern 4231 in a permuta....

The number of successions (or small ascents) of a permutation.

The disorder of a permutation.

The length of the longest pattern of the form k 1 2.

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation....

The number of permutations obtained by switching adjacencies or successions.

The rix statistic of a permutation.

The major index minus the number of excedences of a permutation.

The number of admissible inversions of a permutation.

The number of runs in a permutation.

The sum of the ascent tops of a permutation.

The sum of the ascent bottoms of a permutation.

The number of times a permutation switches from increasing to decreasing or decre....

The sum of St000483 over all subsequences of length at least three.

The length of the longest cycle of a permutation.

The number of cycles of length at least 3 of a permutation.

The length of the shortest cycle of a permutation.

The number of cycle of a permutation of length at most 2.

The number of cycle of a permutation of length at most 3.

The number of inversions of distance at most 3 of a permutation.

The number of inversions of distance at most 2 of a permutation.

Eigenvalues of the random-to-random operator acting on the regular representation....

The size of the first part in the decomposition of a permutation.

The number of stretching pairs of a permutation.

The number of subsequences of a permutation which are not order-isomorphic.

The number of permutations with the same descent word as the given permutation.

The number of 2-rises of a permutation.

The number of even inversions of a permutation.

The number of odd inversions of a permutation.

The number of indices greater than or equal to 2 of a permutation such that all s....

The number of left-to-right-minima of a permutation.

The number of parabolic double cosets with minimal element being the given permut....

The number of global descents of a permutation.

The Edelman-Greene number of a permutation.

The inversion index of a permutation.

The number of cyclic descents of a permutation.

The number of occurrences of the patterns 2143 or 4231 in a permutation.

The number of occurrences of the pattern 52341 in a permutation.

The normalized sum of the minimal distances to a greater element.

The sum of the minimal distances to a greater element.

The number of up-down runs of a permutation.

The number of big ascents of a permutation.

The number of big descents of a permutation.

The number of 2-excedences of a permutation.

The number of 3-excedences of a permutation.

The number of 3-rises of a permutation.

The maximal size of a rise in a permutation.

The maximal difference between successive positions of a permutation.

The last descent of a permutation.

The first descent of a permutation.

The staircase size of the code of a permutation.

The number of right floats of a permutation.

The number of right ropes of a permutation.

The number of rafts of a permutation.

The number of right tethers of a permutation.

The number of permutations obtained by switching ascents or descents of size 2.

The reversal length of a permutation.

The number of minimal elements in Bruhat order not less than the permutation.

The size of the support of a permutation.

The standardized bi-alternating inversion number of a permutation.

The size of the conjugacy class of a permutation.

Babson and SteingrÃmsson's statistic stat of a permutation.

The number of affine bounded permutations that project to a given permutation.

The number of cycles in the breakpoint graph of a permutation.

The number of weak deficiencies of a permutation.

The number of deficiencies of a permutation.

The number of occurrences of 14-2-3 or 14-3-2.

The number of big deficiencies of a permutation.

The number of big exceedences of a permutation.

The label of the leaf of the path following the smaller label in the increasing b....

The smallest label of a leaf of the increasing binary tree associated to a permut....

The normalized sum of the leaf labels of the increasing binary tree associated to....

The largest label of a leaf in the binary search tree associated with the permuta....

The number of double exceedences of a permutation.

The number of double deficiencies of a permutation.

The last entry of a permutation.

The number of big ascents of a permutation after adding the value $\pi(0) = 0$.

The number of occurrences of the pattern 4213 in a permutation.

The number of occurrences of either of the pattern 2143 or 2143 in a permutation.....

The sum of the positions of the left to right maxima of a permutation.

The tier of a permutation.

The mak of a permutation.

The mad of a permutation.

The stat' of a permutation.

The stat`` of a permutation.

The makl of a permutation.

The number of occurrences of the vincular pattern |213 in a permutation.

The number of occurrences of the vincular pattern |231 in a permutation.

The number of occurrences of the vincular pattern |312 in a permutation.

The number of occurrences of the vincular pattern |321 in a permutation.

The number of occurrences of the vincular pattern |132 in a permutation.

The number of occurrences of the vincular pattern |123 in a permutation.

The reduced reflection length of the permutation.

The sum of the number of descents and the number of recoils of a permutation.

The sum of the number of major and the inverse major index of a permutation.

The spearman's rho of a permutation and the identity permutation.

The Ulam distance of a permutation to the identity permutation.

The total displacement of a permutation.

The number of indices that are either descents or recoils.

The number of permutations obtained by reversing blocks of three consecutive numb....

The comajor index of a permutation.

The number of right outer peaks of a permutation.

The number of descents of distance 2 of a permutation.

The number of ascents of distance 2 of a permutation.

The breath of a permutation.

The size of the largest block in the direct sum decomposition of a permutation.

**Posets**(39 statistics) # graph like objects

The number of minimal elements in a poset.

The number of maximal elements of a poset.

The number of antichains in a poset.

The number of maximal chains in a poset.

The rank of the poset.

The number of linear extensions of a poset.

The number of facets in the order polytope of this poset.

The number of facets in the chain polytope of the poset.

The number of chains of a poset.

The number of connected components of the Hasse diagram for the poset.

The number of elements in the poset.

The size of the preimage of the map 'to poset' from Binary trees to Posets.

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

The order dimension or Dushnik-Miller dimension of a poset.

The number of rowmotion orbits of a poset.

The number of cover relations in a poset.

The number of posets with the same order polynomial.

The number of posets with the same zeta polynomial.

The number of posets with combinatorially isomorphic order polytopes.

The width of the poset.

The height of a poset.

The number of modular elements of a lattice.

The number of left modular elements of a lattice.

The jump number of the poset.

The size of the automorphism group of a poset.

The number of endomorphisms of a poset.

The number of strictly order preserving maps of a poset into itself.

The number of relations in a poset.

The rank of the largest boolean interval in a poset.

The number of non-empty boolean intervals in a poset.

The size of the smallest orbit of antichains under Panyushev complementation.

The size of the largest orbit of antichains under Panyushev complementation.

The number of cuts of a poset.

The Grundy value for Hackendot on posets.

The number of ordinal summands of a poset.

The maximal number of elements covered by an element in a poset.

The maximal number of elements covering an element of a poset.

The balance constant multiplied with the number of linear extensions of a poset.

The number of 1/3-balanced pairs in a poset.

**Semistandard tableaux**(11 statistics) # tableau like objects

The cocharge of a semistandard tableau.

The charge of a semistandard tableau.

The sum of the entries of a semistandard tableau.

The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....

The major index of a semistandard tableau obtained by standardizing.

The trace of a semistandard tableau.

The segment statistic of a semistandard tableau.

The flush statistic of a semistandard tableau.

The last entry in the first row of a semistandard tableau.

The last entry on the main diagonal of a semistandard tableau.

The first entry in the last row of a semistandard tableau.

**Set partitions**(92 statistics) # partition like objects

The number of blocks in the set partition.

The size of the orbit of the set partition under rotation.

The rank of the set partition.

Sum of the difference between the maximal and the minimal elements of the blocks ....

Sum of the minimal elements of the blocks of a set partition.

Sum of the maximal elements of the blocks of a set partition.

The number of crossings of a set partition.

The number of nestings of a set partition.

The number of singleton blocks of a set partition.

The number of anti-singletons of a set partition.

The number of nonsingleton blocks of a set partition.

The crossing number of a set partition.

The nesting number of a set partition.

The intertwining number of a set partition.

The number of inversions of a set partition.

The rob statistic of a set partition.

The los statistic of a set partition.

The rcs statistic of a set partition.

The lcb statistic of a set partition.

The lcs statistic of a set partition.

The rcb statistic of a set partition.

The number of successions of a set partitions.

The maximal difference between two elements in a common block.

The cardinality of the first block of a set partition.

The biggest entry in the block containing the 1.

The number of occurrences of the pattern {{1,2},{3}} in a set partition.

The number of occurrences of the pattern {{1,3},{2}} in a set partition.

The number of occurrences of the pattern {{1},{2,3}} in a set partition.

The number of occurrences of the pattern {{1},{2},{3}} in a set partition.

The number of occurrences of the pattern {{1,2}} in a set partition.

The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.

The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.

The number of occurrences of the pattern {{1,2,3}} in a set partition.

The number of internal points of a set partition.

The number of overlapping pairs of blocks of a set partition.

The number of occurrences of the pattern {{1},{2}} in a set partition.

The major index of a set partition.

The dimension exponent of a set partition.

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....

The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.

The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1,2 ar....

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.

The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....

The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.

The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....

The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....

The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....

The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.

The number of blocks in the first part of the atomic decomposition of a set parti....

The dimension of a set partition.

The minimal arc length of a set partition.

The maximal arc length of a set partition.

A variant of the major index of a set partition.

The major index of the permutation obtained by flattening the set partition.

The length of the longest partition in the vacillating tableau corresponding to a....

The number of unsplittable factors of the set partition.

The largest opener of a set partition.

**Standard tableaux**(20 statistics) # tableau like objects

The charge of a standard tableau.

The number of attacking pairs of a standard tableau.

The number of inversions of a standard tableau.

The Shynar inversion number of a standard tableau.

The inversion number of a standard tableau as defined by Haglund and Stevens.

The orbit size of a standard tableau under promotion.

The number of descents of a standard tableau.

The cocharge of a standard tableau.

The (standard) major index of a standard tableau.

The leg major index of a standard tableau.

The number of ascents of a standard tableau.

Eigenvalues of the random-to-random operator acting on a simple module.

The modular (standard) major index of a standard tableau.

The row containing the largest entry of a standard tableau.

The last entry in the first row of a standard tableau.

The last entry on the main diagonal of a standard tableau.

The first entry in the last row of a standard tableau.

The number of entries in a standard Young tableau such that the next integer is a....

The length of the path to the largest entry in a standard Young tableau.

The index of the last row whose first entry is the row number in a standard Young....