Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[1,0] => [2,1] => [1,1] => [1,0,1,0] => 1
[1,0,1,0] => [3,1,2] => [1,2] => [1,0,1,1,0,0] => 2
[1,1,0,0] => [2,3,1] => [2,1] => [1,1,0,0,1,0] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,0] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4
>>> Load all 127 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!