- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000940: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].

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.

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

inner shape

Description

The inner shape of a skew partition.