- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000091: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,5]] => [1,2] => [2] => [2] => 0

[[2,5]] => [1,2] => [2] => [2] => 0

[[3,5]] => [1,2] => [2] => [2] => 0

[[4,5]] => [1,2] => [2] => [2] => 0

[[5,5]] => [1,2] => [2] => [2] => 0

[[1],[5]] => [2,1] => [1,1] => [1,1] => 0

[[2],[5]] => [2,1] => [1,1] => [1,1] => 0

[[3],[5]] => [2,1] => [1,1] => [1,1] => 0

[[4],[5]] => [2,1] => [1,1] => [1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The descent variation of a composition.
Defined in [1].

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.

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).

Map

rotate back to front

Description

The back to front rotation of an integer composition.