- FindStat

Your data matches 248 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000075
(load all 18 compositions to match this statistic)

Mapping

St000075: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => 1
[[1],[2]]
 => 1
[[1,2,3]]
 => 1
[[1,3],[2]]
 => 2
[[1,2],[3]]
 => 2
[[1],[2],[3]]
 => 1
[[1,2,3,4]]
 => 1
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 3
[[1,2,3],[4]]
 => 3
[[1,3],[2,4]]
 => 2
[[1,2],[3,4]]
 => 2
[[1,4],[2],[3]]
 => 3
[[1,3],[2],[4]]
 => 3
[[1,2],[3],[4]]
 => 3
[[1],[2],[3],[4]]
 => 1

Description

The orbit size of a standard tableau under promotion.

Matching statistic: St000003
(load all 3 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [2]
 => 1
[[1],[2]]
 => [1,1]
 => 1
[[1,2,3]]
 => [3]
 => 1
[[1,3],[2]]
 => [2,1]
 => 2
[[1,2],[3]]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1
[[1,2,3,4]]
 => [4]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => 3
[[1,2,4],[3]]
 => [3,1]
 => 3
[[1,2,3],[4]]
 => [3,1]
 => 3
[[1,3],[2,4]]
 => [2,2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => 3
[[1,3],[2],[4]]
 => [2,1,1]
 => 3
[[1,2],[3],[4]]
 => [2,1,1]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 1

Description

The number of [[/StandardTableaux|standard Young tableaux]] of the partition.

Matching statistic: St001415
(load all 16 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
St001415: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => 0 => 1
[[1],[2]]
 => 1 => 1
[[1,2,3]]
 => 00 => 2
[[1,3],[2]]
 => 10 => 1
[[1,2],[3]]
 => 01 => 1
[[1],[2],[3]]
 => 11 => 2
[[1,2,3,4]]
 => 000 => 3
[[1,3,4],[2]]
 => 100 => 1
[[1,2,4],[3]]
 => 010 => 3
[[1,2,3],[4]]
 => 001 => 2
[[1,3],[2,4]]
 => 101 => 3
[[1,2],[3,4]]
 => 010 => 3
[[1,4],[2],[3]]
 => 110 => 2
[[1,3],[2],[4]]
 => 101 => 3
[[1,2],[3],[4]]
 => 011 => 1
[[1],[2],[3],[4]]
 => 111 => 3

Description

The length of the longest palindromic prefix of a binary word.

Matching statistic: St001686
(load all 3 compositions to match this statistic)

Mapping

Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [[2,0],[1]]
 => 1
[[1],[2]]
 => [[1,1],[1]]
 => 1
[[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 1
[[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => 2
[[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 2
[[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 1
[[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 1
[[1,3,4],[2]]
 => [[3,1,0,0],[2,1,0],[1,1],[1]]
 => 3
[[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => 3
[[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => 2
[[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[[1,4],[2],[3]]
 => [[2,1,1,0],[1,1,1],[1,1],[1]]
 => 3
[[1,3],[2],[4]]
 => [[2,1,1,0],[2,1,0],[1,1],[1]]
 => 3
[[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 3
[[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => 1

Description

The order of promotion on a Gelfand-Tsetlin pattern.

Matching statistic: St001780
(load all 3 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [2]
 => 1
[[1],[2]]
 => [1,1]
 => 1
[[1,2,3]]
 => [3]
 => 1
[[1,3],[2]]
 => [2,1]
 => 2
[[1,2],[3]]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1
[[1,2,3,4]]
 => [4]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => 3
[[1,2,4],[3]]
 => [3,1]
 => 3
[[1,2,3],[4]]
 => [3,1]
 => 3
[[1,3],[2,4]]
 => [2,2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => 3
[[1,3],[2],[4]]
 => [2,1,1]
 => 3
[[1,2],[3],[4]]
 => [2,1,1]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 1

Description

The order of promotion on the set of standard tableaux of given shape.

Matching statistic: St000734
(load all 2 compositions to match this statistic)

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [[1,2]]
 => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1

Description

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

Matching statistic: St000503
(load all 2 compositions to match this statistic)

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1],[2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2],[3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1

Description

The maximal difference between two elements in a common block.

Matching statistic: St000626
(load all 18 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => 0 => 0 => 1
[[1],[2]]
 => 1 => 1 => 1
[[1,2,3]]
 => 00 => 10 => 2
[[1,3],[2]]
 => 10 => 01 => 2
[[1,2],[3]]
 => 01 => 00 => 1
[[1],[2],[3]]
 => 11 => 11 => 1
[[1,2,3,4]]
 => 000 => 010 => 2
[[1,3,4],[2]]
 => 100 => 101 => 2
[[1,2,4],[3]]
 => 010 => 100 => 3
[[1,2,3],[4]]
 => 001 => 110 => 3
[[1,3],[2,4]]
 => 101 => 001 => 3
[[1,2],[3,4]]
 => 010 => 100 => 3
[[1,4],[2],[3]]
 => 110 => 011 => 3
[[1,3],[2],[4]]
 => 101 => 001 => 3
[[1,2],[3],[4]]
 => 011 => 000 => 1
[[1],[2],[3],[4]]
 => 111 => 111 => 1

Description

The minimal period of a binary word. This is the smallest natural number

$p$ such that

$w_i=w_{i+p}$ for all

$i\in\{1,\dots,|w|-p\}$ .

Matching statistic: St000638
(load all 3 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000638: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [1,2] => [1] => 1
[[1],[2]]
 => [2,1] => [1] => 1
[[1,2,3]]
 => [1,2,3] => [1,2] => 1
[[1,3],[2]]
 => [2,1,3] => [2,1] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => 2
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [2,1,3] => 3
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,2] => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,1,3] => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1,2] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => 2

Description

The number of up-down runs of a permutation. An '''up-down run''' of a permutation

$\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or

$\pi_{1}$ if 1 is a descent of

$\pi$ . For example, the up-down runs of

$\pi=85712643$ are

$8$ ,

$85$ ,

$57$ ,

$71$ ,

$126$ , and

$643$ .

Matching statistic: St000728
(load all 2 compositions to match this statistic)

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1],[2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2],[3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1

Description

The dimension of a set partition. This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks. A slightly shifted definition of the dimension is [[St000572]].

The following 238 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001246The maximal difference between two consecutive entries of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001595The number of standard Young tableaux of the skew partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000057The Shynar inversion number of a standard tableau. St000064The number of one-box pattern of a permutation. St000178Number of free entries. St000222The number of alignments in the permutation. St000293The number of inversions of a binary word. St000505The biggest entry in the block containing the 1. St000538The number of even inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000836The number of descents of distance 2 of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001742The difference of the maximal and the minimal degree in a graph. St000001The number of reduced words for a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000047The number of standard immaculate tableaux of a given shape. St000054The first entry of the permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000078The number of alternating sign matrices whose left key is the permutation. St000100The number of linear extensions of a poset. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000246The number of non-inversions of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000316The number of non-left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000327The number of cover relations in a poset. St000339The maf index of a permutation. St000378The diagonal inversion number of an integer partition. St000452The number of distinct eigenvalues of a graph. St000485The length of the longest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000722The number of different neighbourhoods in a graph. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000789The number of crossing-similar perfect matchings of a perfect matching. St000795The mad of a permutation. St000820The number of compositions obtained by rotating the composition. St000831The number of indices that are either descents or recoils. St000844The size of the largest block in the direct sum decomposition of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001090The number of pop-stack-sorts needed to sort a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001313The number of Dyck paths above the lattice path given by a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001725The harmonious chromatic number of a graph. St001726The number of visible inversions of a permutation. St001779The order of promotion on the set of linear extensions of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000018The number of inversions of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000089The absolute variation of a composition. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000216The absolute length of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000235The number of indices that are not cyclical small weak excedances. St000290The major index of a binary word. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000369The dinv deficit of a Dyck path. 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. St000381The largest part of an integer composition. St000382The first part of an integer composition. 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. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000463The number of admissible inversions of a permutation. St000471The sum of the ascent tops of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000507The number of ascents of a standard tableau. St000572The dimension exponent of a set partition. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000708The product of the parts of an integer partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000736The last entry in the first row of a semistandard tableau. St000747A variant of the major index of a set partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000808The number of up steps of the associated bargraph. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. St000839The largest opener of a set partition. St000849The number of 1/3-balanced pairs in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001120The length of a longest path in 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. St001279The sum of the parts of an integer partition that are at least two. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001403The number of vertical separators in a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001462The number of factors of a standard tableaux under concatenation. St001485The modular major index of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001497The position of the largest weak excedence of a permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. 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. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001727The number of invisible inversions of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001869The maximum cut size of a graph. St001959The product of the heights of the peaks of a Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000393The number of strictly increasing runs in a binary word. St000981The length of the longest zigzag subpath. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000045The number of linear extensions of a binary tree. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000848The balance constant multiplied with the number of linear extensions of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001118The acyclic chromatic index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000456The monochromatic index of a connected graph. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001623The number of doubly irreducible elements of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000522The number of 1-protected nodes of a rooted tree. St001095The number of non-isomorphic posets with precisely one further covering relation. St000521The number of distinct subtrees of an ordered tree. St000906The length of the shortest maximal chain in a poset. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. 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. St001060The distinguishing index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000166The depth minus 1 of an ordered tree. St001857The number of edges in the reduced word graph of a signed permutation. St000094The depth of an ordered tree. St000260The radius of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St000080The rank of the poset. St000264The girth of a graph, which is not a tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph.