- FindStat

Identifier

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000284: Integer partitions ⟶ ℤ

Values

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

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

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

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

[1,2,3,5,4] => [3,2] => [[4,3],[2]] => [2] => 1

[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[2,1,3,5,4] => [3,2] => [[4,3],[2]] => [2] => 1

[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[2,3,1,5,4] => [3,2] => [[4,3],[2]] => [2] => 1

[2,3,5,4,1] => [4,1] => [[4,4],[3]] => [3] => 1

[2,4,1,5,3] => [3,2] => [[4,3],[2]] => [2] => 1

[2,4,3,1,5] => [3,2] => [[4,3],[2]] => [2] => 1

[2,4,3,5,1] => [3,2] => [[4,3],[2]] => [2] => 1

[2,4,5,3,1] => [4,1] => [[4,4],[3]] => [3] => 1

[2,5,1,4,3] => [4,1] => [[4,4],[3]] => [3] => 1

[2,5,3,1,4] => [3,2] => [[4,3],[2]] => [2] => 1

[2,5,3,4,1] => [4,1] => [[4,4],[3]] => [3] => 1

[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[3,1,2,5,4] => [3,2] => [[4,3],[2]] => [2] => 1

[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[3,2,1,5,4] => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1

[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[3,4,1,5,2] => [3,2] => [[4,3],[2]] => [2] => 1

[3,4,2,1,5] => [3,2] => [[4,3],[2]] => [2] => 1

[3,4,2,5,1] => [3,2] => [[4,3],[2]] => [2] => 1

[3,4,5,2,1] => [4,1] => [[4,4],[3]] => [3] => 1

[3,5,1,4,2] => [4,1] => [[4,4],[3]] => [3] => 1

[3,5,2,1,4] => [3,2] => [[4,3],[2]] => [2] => 1

[3,5,2,4,1] => [4,1] => [[4,4],[3]] => [3] => 1

[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[4,1,2,5,3] => [3,2] => [[4,3],[2]] => [2] => 1

[4,1,3,2,5] => [3,2] => [[4,3],[2]] => [2] => 1

[4,1,3,5,2] => [3,2] => [[4,3],[2]] => [2] => 1

[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[4,2,1,5,3] => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1

[4,2,3,1,5] => [3,2] => [[4,3],[2]] => [2] => 1

[4,2,3,5,1] => [3,2] => [[4,3],[2]] => [2] => 1

[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[4,5,1,3,2] => [4,1] => [[4,4],[3]] => [3] => 1

[4,5,2,1,3] => [3,2] => [[4,3],[2]] => [2] => 1

[4,5,2,3,1] => [4,1] => [[4,4],[3]] => [3] => 1

[4,5,3,1,2] => [3,2] => [[4,3],[2]] => [2] => 1

[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 4

[5,1,2,4,3] => [4,1] => [[4,4],[3]] => [3] => 1

[5,1,3,2,4] => [3,2] => [[4,3],[2]] => [2] => 1

[5,1,3,4,2] => [4,1] => [[4,4],[3]] => [3] => 1

[5,1,4,3,2] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[5,2,1,4,3] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[5,2,3,1,4] => [3,2] => [[4,3],[2]] => [2] => 1

[5,2,3,4,1] => [4,1] => [[4,4],[3]] => [3] => 1

[5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 4

[5,3,1,4,2] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[5,3,2,4,1] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[5,3,4,2,1] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[5,4,1,3,2] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[5,4,2,3,1] => [1,3,1] => [[3,3,1],[2]] => [2] => 1

[5,4,3,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1

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

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

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

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

[1,2,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 25

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

[1,2,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 9

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

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

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

[1,2,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 9

[1,2,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 9

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

[1,2,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 9

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

[1,2,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.

Map

DEX composition

Description

The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.

Map

inner shape

Description

The inner shape of a skew partition.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.