- FindStat

Identifier

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000478: 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] => 0

[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] => 0

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

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

[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] => 2

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

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

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

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

[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] => 0

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

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

[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] => 2

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

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

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

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

[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] => 0

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

[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] => 0

[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] => 2

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

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

[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] => -1

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

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

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

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

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

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

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

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

[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] => 0

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

[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] => 2

[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] => 2

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

[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] => 2

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

[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] => 0

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

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

[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] => 2

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

[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] => 2

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searching the database for the individual values of this statistic

Description

Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.

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.