- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5913 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + 4\ q$

$F_{4} = 8 + 11\ q + 5\ q^{2}$

$F_{5} = 33 + 44\ q + 43\ q^{2}$

$F_{6} = 142 + 287\ q + 230\ q^{2} + 61\ q^{3}$

$F_{7} = 761 + 1904\ q + 1469\ q^{2} + 906\ q^{3}$

Description

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module

$M$ over an algebra

$A$ k-regular in case it has projective dimension k and

$Ext_A^i(M,A)=0$ for

$i \neq k$ and

$Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by OEIS:A035929 .

Map

bounce path

Description

The bounce path determined by an integer composition.

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

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .