- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000619: Permutations ⟶ ℤ

Values

[1,0] => [2,1] => [1,2] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [1,2,3] => [2,3,1] => 1
[1,1,0,0] => [2,3,1] => [1,2,3] => [2,3,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [1,4,2,3] => [3,4,2,1] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [1,3,2,4] => [3,2,4,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,5,2,4,3] => [3,5,4,2,1] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,4,2,3,5] => [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,4,5,2,3] => [4,5,2,3,1] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,5,2,4,3] => [3,5,4,2,1] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,2,4,3,5] => [2,4,3,5,1] => 2
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,5,2,3,4] => [3,4,5,2,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,3,2,5,4] => [3,2,5,4,1] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,2,6,3,5,4] => [2,4,6,5,3,1] => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,2,5,3,4,6] => [2,4,5,3,6,1] => 2
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,2,5,6,3,4] => [2,5,6,3,4,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,6,2,4,5,3] => [3,6,4,5,2,1] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,4,6,2,5,3] => [4,6,2,5,3,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,5,2,3,6] => [4,5,2,3,6,1] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,4,5,6,2,3] => [5,6,2,3,4,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [5,2,3,4,6,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,3,6,2,5,4] => [4,2,6,5,3,1] => 4
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,5,6,2,4,3] => [4,6,5,2,3,1] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [2,5,3,4,6,1] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,2,6,3,4,5] => [2,4,5,6,3,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 2
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,2,6,3,4,5] => [2,4,5,6,3,1] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,5,6,2,3,4] => [4,5,6,2,3,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,4,5,2,3,6] => [4,5,2,3,6,1] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,6,2,3,5,4] => [3,4,6,5,2,1] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,3,5,2,6,4] => [4,2,6,3,5,1] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,3,4,2,6,5] => [4,2,3,6,5,1] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => [2,4,5,3,6,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => [2,4,3,5,6,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,4,2,3,6,5] => [3,4,2,6,5,1] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1

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

$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 4\ q + 7\ q^{2} + 3\ q^{3}$

$F_{5} = 5\ q + 21\ q^{2} + 15\ q^{3} + q^{4}$

Description

The number of cyclic descents of a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is given by the number of indices

$1 \leq i \leq n$ such that

$\pi(i) > \pi(i+1)$ where we set

$\pi(n+1) = \pi(1)$ .

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

Inverse Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$c\pi^{-1}$ where

$c = (1,\ldots,n)$ is the long cycle.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.