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Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤ (values match St000932The number of occurrences of the pattern UDU in a Dyck path., St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.)

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] => 0

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

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

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

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

[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] => 0

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

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

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

[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] => 1

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

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

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

[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] => 1

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

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

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

[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] => 1

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

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