Your data matches 277 different statistics following compositions of up to 3 maps.
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St000784: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 3
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 5
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000209: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
Description
Maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the maximum of this value over all cycles in the permutation.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St000519: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
Description
The largest length of a factor maximising the subword complexity. Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal: $$ H_w = \max\{n: p_w(n)\text{ is maximal}\} $$ A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St000922: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
Description
The minimal number such that all substrings of this length are unique.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000956: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
Description
The maximal displacement of a permutation. This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$. This statistic without the absolute value is the maximal drop size [[St000141]].
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St000982: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
Description
The length of the longest constant subword.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St001416: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
Description
The length of a longest palindromic factor of a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St001417: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
Description
The length of a longest palindromic subword of a binary word. A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword. Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001566: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [1,2] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
Description
The length of the longest arithmetic progression in a permutation. For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with $$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00155: Standard tableaux promotionStandard tableaux
St000743: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [[1]]
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [[1,2]]
=> 1 = 2 - 1
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 2 = 3 - 1
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[3,1]
=> [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> 2 = 3 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 3 = 4 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> 2 = 3 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
Description
The number of entries in a standard Young tableau such that the next integer is a neighbour.
The following 267 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000806The semiperimeter of the associated bargraph. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000060The greater neighbor of the maximum. St000144The pyramid weight of the Dyck path. St000171The degree of the graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000381The largest part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000503The maximal difference between two elements in a common block. St000626The minimal period of a binary word. St000702The number of weak deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000983The length of the longest alternating subword. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001117The game chromatic index of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000141The maximum drop size of a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000459The hook length of the base cell of a partition. St000505The biggest entry in the block containing the 1. St000672The number of minimal elements in Bruhat order not less than the permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000837The number of ascents of distance 2 of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001405The number of bonds in a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001948The number of augmented double ascents of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000295The length of the border of a binary word. St000836The number of descents of distance 2 of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000907The number of maximal antichains of minimal length in a poset. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000216The absolute length of a permutation. St000831The number of indices that are either descents or recoils. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000390The number of runs of ones in a binary word. St001267The length of the Lyndon factorization of the binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000393The number of strictly increasing runs in a binary word. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000259The diameter of a connected graph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000520The number of patterns in a permutation. St000522The number of 1-protected nodes of a rooted tree. St000643The size of the largest orbit of antichains under Panyushev complementation. St000521The number of distinct subtrees of an ordered tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000834The number of right outer peaks of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000264The girth of a graph, which is not a tree. St000308The height of the tree associated to a permutation. St000438The position of the last up step in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001517The length of a longest pair of twins in a permutation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001838The number of nonempty primitive factors of a binary word. St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. 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. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000735The last entry on the main diagonal of a standard tableau. St001645The pebbling number of a connected graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000245The number of ascents of a permutation. St001060The distinguishing index of a graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000222The number of alignments in the permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000989The number of final rises of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001388The number of non-attacking neighbors of a permutation. St001535The number of cyclic alignments of a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St000004The major index of a permutation. St000021The number of descents of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000062The length of the longest increasing subsequence of the permutation. St000080The rank of the poset. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000653The last descent of a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000991The number of right-to-left minima of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001220The width of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence 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. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001874Lusztig's a-function for the symmetric group. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000528The height of a poset. St000619The number of cyclic descents of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000906The length of the shortest maximal chain in a poset. St001375The pancake length of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001782The order of rowmotion on the set of order ideals of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000910The number of maximal chains of minimal length in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001902The number of potential covers of a poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000739The first entry in the last row of a semistandard tableau. St000762The sum of the positions of the weak records of 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. St000978The sum of the positions of double down-steps of a Dyck path. St001118The acyclic chromatic index of a graph. St001401The number of distinct entries in a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.