- FindStat

Your data matches 206 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000075
(load all 17 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]]
 => 1 = 0 + 1
[[1,2]]
 => 1 = 0 + 1
[[1],[2]]
 => 1 = 0 + 1
[[1,2,3]]
 => 1 = 0 + 1
[[1,3],[2]]
 => 2 = 1 + 1
[[1,2],[3]]
 => 2 = 1 + 1
[[1],[2],[3]]
 => 1 = 0 + 1
[[1,2,3,4]]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => 3 = 2 + 1
[[1,3],[2],[4]]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => 1 = 0 + 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]]
 => [1]
 => 1 = 0 + 1
[[1,2]]
 => [2]
 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => 1 = 0 + 1
[[1,2,3]]
 => [3]
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [3,1]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [3,1]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => [2,2]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [2,2]
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 3 = 2 + 1
[[1,3],[2],[4]]
 => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [2,1,1]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 1 = 0 + 1

Description

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

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

Description

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

Matching statistic: St000057
(load all 2 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.

Matching statistic: St000178
(load all 2 compositions to match this statistic)

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000178: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of free entries. The ''tiling'' of a pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called ''tiles'', and each entry in a pattern belong to exactly one tile. A tile is ''free'' if it do not intersect any of the first and the last row. This statistic is the total number of entries that belong to a free tile.

Matching statistic: St000222
(load all 32 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of alignments in the permutation.

Matching statistic: St001742
(load all 2 compositions to match this statistic)

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001742: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The difference of the maximal and the minimal degree in a graph. The graph is regular if and only if this statistic is zero.

Matching statistic: St001595
(load all 3 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of standard Young tableaux of the skew partition.

Matching statistic: St001929
(load all 6 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001929: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path.

Matching statistic: St000002

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

The following 196 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000089The absolute variation of a composition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000209Maximum difference of elements in cycles. 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. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. 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. 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. 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. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000670The reversal length of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000921The number of internal inversions of a binary word. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. 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. St000001The number of reduced words for a permutation. St000047The number of standard immaculate tableaux of a given shape. St000058The order of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. 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. St000638The number of up-down runs of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000820The number of compositions obtained by rotating the composition. St000883The number of longest increasing subsequences of a permutation. 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. 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. St001779The order of promotion on the set of linear extensions of a poset. St001415The length of the longest palindromic prefix of a binary word. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000293The number of inversions of a binary word. St000538The number of even inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. 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. St000503The maximal difference between two elements in a common block. St000626The minimal period of a binary word. St000728The dimension of a set partition. 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. St000064The number of one-box pattern of a permutation. 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. St000171The degree of the graph. St000216The absolute length of a permutation. St000290The major index of a binary word. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000369The dinv deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. 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. St000651The maximal size of a rise in a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000747A variant of the major index of a set 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. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. 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. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001120The length of a longest path in a graph. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001485The modular major index of a binary word. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001869The maximum cut size of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000054The first entry of the permutation. St000060The greater neighbor of the maximum. St000100The number of linear extensions of a poset. St000240The number of indices that are not small excedances. 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. 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. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000722The number of different neighbourhoods in a graph. St000795The mad of a permutation. 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. 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. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St000471The sum of the ascent tops of a permutation. St000708The product of the parts of an integer partition. St001959The product of the heights of the peaks of a Dyck path. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. 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,. St000045The number of linear extensions of a binary tree. St000848The balance constant multiplied with the number of linear extensions of a poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St001280The number of parts of an integer partition that are at least two. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001857The number of edges in the reduced word graph of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001722The number of minimal chains with small intervals between a binary word and the top element. St000456The monochromatic index of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000102The charge of a semistandard tableau. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. 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. St000997The even-odd crank of an integer partition. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.