- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000031: Permutations ⟶ ℤ

Values

[[1,2]] => [1,2] => [1,2] => [1,2] => 2

[[2,2]] => [1,2] => [1,2] => [1,2] => 2

[[1],[2]] => [2,1] => [2,1] => [1,2] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,5]] => [1,2] => [1,2] => [1,2] => 2

[[2,5]] => [1,2] => [1,2] => [1,2] => 2

[[3,5]] => [1,2] => [1,2] => [1,2] => 2

[[4,5]] => [1,2] => [1,2] => [1,2] => 2

[[5,5]] => [1,2] => [1,2] => [1,2] => 2

[[1],[5]] => [2,1] => [2,1] => [1,2] => 2

[[2],[5]] => [2,1] => [2,1] => [1,2] => 2

[[3],[5]] => [2,1] => [2,1] => [1,2] => 2

[[4],[5]] => [2,1] => [2,1] => [1,2] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2321 entries. <<<

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Description

The number of cycles in the cycle decomposition of a permutation.

Map

toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).