- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The length of the partition.

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

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.