- FindStat

Your data matches 166 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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
[[1,2,3],[4]]
 => [3,1]
 => 3 = 2 + 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
[[1,2,3,4,5]]
 => [5]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => 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
[[1,2,3],[4]]
 => [3,1]
 => 3 = 2 + 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
[[1,2,3,4,5]]
 => [5]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => 1 = 0 + 1

Description

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

Matching statistic: St001908
(load all 3 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
St001908: 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
[[1,2,3],[4]]
 => [3,1]
 => 3 = 2 + 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
[[1,2,3,4,5]]
 => [5]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => 1 = 0 + 1

Description

The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. For example, there are eight tableaux of shape

$[3,2,1]$ with maximal entry

$3$ , but two of them have the same weight.

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
[[1,2,3],[4]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[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
[[1,2,3,4,5]]
 => [5]
 => [[1,2,3,4,5]]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[[1,2,3,4,5,6,7]]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 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: St000682
(load all 6 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1]
 => 10 => 0
[[1,2]]
 => [2]
 => 100 => 0
[[1],[2]]
 => [1,1]
 => 110 => 0
[[1,2,3]]
 => [3]
 => 1000 => 0
[[1,3],[2]]
 => [2,1]
 => 1010 => 1
[[1,2],[3]]
 => [2,1]
 => 1010 => 1
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 0
[[1,2,3,4]]
 => [4]
 => 10000 => 0
[[1,3,4],[2]]
 => [3,1]
 => 10010 => 2
[[1,2,3],[4]]
 => [3,1]
 => 10010 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => 10110 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => 10110 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => 10110 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 11110 => 0
[[1,2,3,4,5]]
 => [5]
 => 100000 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 111110 => 0
[[1,2,3,4,5,6]]
 => [6]
 => 1000000 => 0
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[[1,2,3,4,5,6,7]]
 => [7]
 => 10000000 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => 11111110 => 0

Description

The Grundy value of Welter's game on a binary word. Two players take turns moving a

$1$ to the left. The loosing positions are the words

$1\dots 10\dots 0$ .

Matching statistic: St001698
(load all 2 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [[1]]
 => 0
[[1,2]]
 => [1,2] => [[1,2]]
 => 0
[[1],[2]]
 => [2,1] => [[1],[2]]
 => 0
[[1,2,3]]
 => [1,2,3] => [[1,2,3]]
 => 0
[[1,3],[2]]
 => [2,1,3] => [[1,3],[2]]
 => 1
[[1,2],[3]]
 => [3,1,2] => [[1,3],[2]]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [[1],[2],[3]]
 => 0
[[1,2,3,4]]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [[1,4],[2],[3]]
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [[1,4],[2],[3]]
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
 => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7]]
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7]]
 => 0

Description

The comajor index of a standard tableau minus the weighted size of its shape.

Matching statistic: St001699
(load all 2 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001699: 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
[[1,2,3],[4]]
 => [3,1]
 => [[1,2,3],[4]]
 => 2
[[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
[[1,2,3,4,5]]
 => [5]
 => [[1,2,3,4,5]]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[[1,2,3,4,5,6,7]]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 0

Description

The major index of a standard tableau minus the weighted size of its shape.

Matching statistic: St000277
(load all 6 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1] => 1 = 0 + 1
[[1,2]]
 => [1,2] => [2] => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,1] => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [3] => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [4] => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3] => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,3] => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,2] => 3 = 2 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,2] => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [5] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6] => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7] => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => 1 = 0 + 1

Description

The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a

$2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition

$c=c_1,\dots,c_k$ of

$n$ , the number of ribbon shaped standard tableaux equals

$\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},$ where the sum is over all coarsenings of

$c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

Matching statistic: St001313
(load all 6 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1]
 => 10 => 1 = 0 + 1
[[1,2]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => 110 => 1 = 0 + 1
[[1,2,3]]
 => [3]
 => 1000 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,2],[3]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[[1,2,3,4]]
 => [4]
 => 10000 => 1 = 0 + 1
[[1,3,4],[2]]
 => [3,1]
 => 10010 => 3 = 2 + 1
[[1,2,3],[4]]
 => [3,1]
 => 10010 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[[1,3],[2],[4]]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5]
 => 100000 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => 1000000 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => 1111110 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => 10000000 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => 11111110 => 1 = 0 + 1

Description

The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating

$1$ 's as steps

$(1,0)$ and

$0$ 's as steps

$(0,1)$ . Given a binary word

$w$ , this statistic counts the number of lattice paths from the origin to the same endpoint as

$w$ that stay weakly above

$w$ . See [[St001312]] for this statistic on compositions treated as bounce paths.

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
[[1,2,3],[4]]
 => [3,1]
 => [[3,1],[]]
 => 3 = 2 + 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
[[1,2,3,4,5]]
 => [5]
 => [[5],[]]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => [[6],[]]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => [[7],[]]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => 1 = 0 + 1

Description

The number of standard Young tableaux of the skew partition.

The following 156 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000089The absolute variation of a composition. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000293The number of inversions of a binary word. St000377The dinv defect of an integer partition. St000425The number of occurrences of the pattern 132 or of the pattern 213 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. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St001584The area statistic between a Dyck path and its bounce path. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001695The natural comajor index of a standard Young tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000047The number of standard immaculate tableaux of a given shape. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000670The reversal length of a permutation. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. 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. St000075The orbit size of a standard tableau under promotion. St000543The size of the conjugacy class of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000369The dinv deficit of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001910The height of the middle non-run of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000626The minimal period of a binary word. St000290The major index of a binary word. St000355The number of occurrences of the pattern 21-3. St000376The bounce deficit of a Dyck path. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000491The number of inversions of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000747A variant of the major index of a set partition. St001377The major index minus the number of inversions of a permutation. St001485The modular major index of a binary word. St000100The number of linear extensions of a poset. St000548The number of different non-empty partial sums of an integer partition. 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. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St000530The number of permutations with the same descent word as the given permutation. St000222The number of alignments in the permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000799The number of occurrences of the vincular pattern |213 in 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". St000001The number of reduced words for a permutation. St000525The number of posets with the same zeta polynomial. St001246The maximal difference between two consecutive entries of a permutation. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000058The order of a permutation. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000988The orbit size of a permutation under Foata's bijection. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000209Maximum difference of elements in cycles. St000494The number of inversions of distance at most 3 of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001510The number of self-evacuating linear extensions of a finite poset. 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. St001760The number of prefix or suffix reversals needed to sort a permutation. 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. St000141The maximum drop size of a permutation. St000485The length of the longest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St000029The depth of a permutation. St000216The absolute length of a permutation. 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. 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. St001480The number of simple summands of the module J^2/J^3. St000045The number of linear extensions of a binary tree. St000219The number of occurrences of the pattern 231 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected 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

$K[x]/(x^n)$ . St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St000102The charge of a semistandard tableau. St000259The diameter of a connected graph. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001890The maximum magnitude of the Möbius function of a poset.