Your data matches 600 different statistics following compositions of up to 3 maps.
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Matching statistic: St000847
(load all 43 compositions to match this statistic)
Mapping
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 1
10 => 1
11 => 1
000 => 1
001 => 1
010 => 2
011 => 1
100 => 1
101 => 2
110 => 1
111 => 1
Description
The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation. For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
Matching statistic: St000921
(load all 21 compositions to match this statistic)
Mapping
St000921: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 0 = 1 - 1
10 => 0 = 1 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 0 = 1 - 1
010 => 0 = 1 - 1
011 => 0 = 1 - 1
100 => 1 = 2 - 1
101 => 0 = 1 - 1
110 => 1 = 2 - 1
111 => 0 = 1 - 1
Description
The number of internal inversions of a binary word. Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted. An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
Matching statistic: St001423
(load all 55 compositions to match this statistic)
Mapping
St001423: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 0 = 1 - 1
10 => 0 = 1 - 1
11 => 0 = 1 - 1
000 => 1 = 2 - 1
001 => 0 = 1 - 1
010 => 0 = 1 - 1
011 => 0 = 1 - 1
100 => 0 = 1 - 1
101 => 0 = 1 - 1
110 => 0 = 1 - 1
111 => 1 = 2 - 1
Description
The number of distinct cubes in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uuu$ is a factor of the word.
Matching statistic: St001730
(load all 40 compositions to match this statistic)
Mapping
St001730: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 0 = 1 - 1
10 => 0 = 1 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 0 = 1 - 1
010 => 0 = 1 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 0 = 1 - 1
110 => 0 = 1 - 1
111 => 0 = 1 - 1
Description
The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St000047
(load all 5 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 2
010 => [1,1,1] => 1
011 => [1,2] => 1
100 => [1,2] => 1
101 => [1,1,1] => 1
110 => [2,1] => 2
111 => [3] => 1
Description
The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000758
(load all 6 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 1
010 => [1,1,1] => 1
011 => [1,2] => 2
100 => [1,2] => 2
101 => [1,1,1] => 1
110 => [2,1] => 1
111 => [3] => 1
Description
The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000760
(load all 5 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 2
010 => [1,1,1] => 1
011 => [1,2] => 1
100 => [1,2] => 1
101 => [1,1,1] => 1
110 => [2,1] => 2
111 => [3] => 1
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000764
(load all 7 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 1
010 => [1,1,1] => 1
011 => [1,2] => 2
100 => [1,2] => 2
101 => [1,1,1] => 1
110 => [2,1] => 1
111 => [3] => 1
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St000816
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 1
1 => [1,1] => 1
00 => [3] => 1
01 => [2,1] => 1
10 => [1,2] => 1
11 => [1,1,1] => 1
000 => [4] => 1
001 => [3,1] => 1
010 => [2,2] => 2
011 => [2,1,1] => 1
100 => [1,3] => 2
101 => [1,2,1] => 1
110 => [1,1,2] => 1
111 => [1,1,1,1] => 1
Description
The number of standard composition tableaux of the composition. See [1, Def. 4.2.6]. Apparently, the total number of tableaux of given size is the number of involutions.
Matching statistic: St000091
(load all 7 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 0 = 1 - 1
1 => [1] => 0 = 1 - 1
00 => [2] => 0 = 1 - 1
01 => [1,1] => 0 = 1 - 1
10 => [1,1] => 0 = 1 - 1
11 => [2] => 0 = 1 - 1
000 => [3] => 0 = 1 - 1
001 => [2,1] => 0 = 1 - 1
010 => [1,1,1] => 0 = 1 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 0 = 1 - 1
110 => [2,1] => 0 = 1 - 1
111 => [3] => 0 = 1 - 1
Description
The descent variation of a composition. Defined in [1].
The following 590 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000079The number of alternating sign matrices for a given Dyck path. St000277The number of ribbon shaped standard tableaux. St000390The number of runs of ones in a binary word. St000767The number of runs in an integer composition. St000785The number of distinct colouring schemes of a graph. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001282The number of graphs with the same chromatic polynomial. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000089The absolute variation of a composition. St000095The number of triangles of a graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000348The non-inversion sum of a binary word. St000386The number of factors DDU in a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001308The number of induced paths on three vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001350Half of the Albertson index of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001656The monophonic position number of a graph. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001736The total number of cycles in a graph. St001764The number of non-convex subsets of vertices in a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001871The number of triconnected components of a graph. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000048The multinomial of the parts of a partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000108The number of partitions contained in the given partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000201The number of leaf nodes in a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000273The domination number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000287The number of connected components of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000346The number of coarsenings of a partition. St000354The number of recoils of a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000396The register function (or Horton-Strahler number) of a binary tree. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000701The protection number of a binary tree. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000862The number of parts of the shifted shape of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000948The chromatic discriminant of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001162The minimum jump of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001432The order dimension of the partition. St001463The number of distinct columns in the nullspace of a graph. St001571The Cartan determinant of the integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001595The number of standard Young tableaux of the skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001616The number of neutral elements in a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001642The Prague dimension of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001675The number of parts equal to the part in the reversed composition. St001679The number of subsets of a lattice whose meet is the bottom element. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001754The number of tolerances of a finite lattice. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001814The number of partitions interlacing the given partition. St001829The common independence number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001917The order of toric promotion on the set of labellings of a graph. St001928The number of non-overlapping descents in a permutation. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000002The number of occurrences of the pattern 123 in a permutation. St000008The major index of the composition. St000010The length of the partition. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000325The width of the tree associated to a permutation. St000347The inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000407The number of occurrences of the pattern 2143 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000459The hook length of the base cell of a partition. St000463The number of admissible inversions of a permutation. St000470The number of runs in a permutation. St000475The number of parts equal to 1 in a partition. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000534The number of 2-rises of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000709The number of occurrences of 14-2-3 or 14-3-2. St000731The number of double exceedences of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000768The number of peaks in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000842The breadth of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000944The 3-degree of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001127The sum of the squares of the parts of a partition. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001377The major index minus the number of inversions of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001411The number of patterns 321 or 3412 in a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. St001521Half the total irregularity of a graph. St001549The number of restricted non-inversions between exceedances. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001856The number of edges in the reduced word graph of a permutation. St001911A descent variant minus the number of inversions. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001128The exponens consonantiae of a partition. St001220The width of a permutation. St001271The competition number of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000064The number of one-box pattern of a permutation. St000353The number of inner valleys of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000747A variant of the major index of a set partition. St000779The tier of 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. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000893The number of distinct diagonal sums of an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. 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. St000961The shifted major index of a permutation. 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. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000260The radius of a connected graph. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001389The number of partitions of the same length below the given integer partition. St000741The Colin de Verdière graph invariant. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001812The biclique partition number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001625The Möbius invariant of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000045The number of linear extensions of a binary tree. St000284The Plancherel distribution on integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001763The Hurwitz number of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000219The number of occurrences of the pattern 231 in a permutation. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001118The acyclic chromatic index of a graph.