- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤ

Values

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

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

[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] => 1

[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] => 1

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

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

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

[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] => 1

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

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

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

[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] => 1

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

[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] => 1

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

[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] => 1

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

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

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

[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] => 1

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

[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] => 1

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

[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] => 1

[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] => 1

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

[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] => 1

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

[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] => 1

[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] => 1

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

[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] => 1

[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] => 1

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

[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] => 1

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

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

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

[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] => 1

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

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

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

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

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

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

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

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Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition

$\lambda \vdash n$ by the number of standard tableaux

$T$ of shape

$\lambda$ such that

$\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..

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

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

inner shape

Description

The inner shape of a skew partition.