- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = 9\ q + 11\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 39\ q + 48\ q^{2} + 28\ q^{3} + 4\ q^{4} + q^{5}$

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

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

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

bounce path

Description

The bounce path determined by an integer composition.