- FindStat

Identifier

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

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

bounce path

Description

The bounce path determined by an integer composition.