- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001188: Dyck paths ⟶ ℤ (values match St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.)

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] => 1

[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] => 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] => 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] => 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] => 2

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

[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] => 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] => 2

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

[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] => 1

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

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

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

[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] => 2

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

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

[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] => 2

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

[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] => 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] => 2

[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] => 1

[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] => 1

[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 873 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

click to show known generating functions

Description

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.
Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.

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