- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001001: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 5 + q^{3}$

$F_{4} = 23 + q^{6}$

$F_{5} = 103 + 16\ q + q^{10}$

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

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

bounce path

Description

The bounce path determined by an integer composition.

Map

Cori-Le Borgne involution

Description

The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite

$\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$ , where

$\zeta$ is Mp00030zeta map.