- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001513: Permutations ⟶ ℤ

Values

[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,4,1,5,2] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,3,4,2] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [4,2,5,1,3] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,3,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,3,2,5,4] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [4,2,5,6,1,3] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,5,6,3,4,2] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,1,5,3] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,2,3,1,4] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,3,6,2,5,4] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,1,6,4,5,2] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,4,2,1,6,3] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,4,6,2,1,5] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,2,1,3,6,5] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,5,2,6,1,4] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,2,5,3,1,4] => 1
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,3,6,7,2,5,4] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,4,1,5,6,2] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,3,4,5,2] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,2,5,1,6,3] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,4,2,6,5] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,3,2,5,6,4] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [5,2,3,6,1,4] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,5,3,4,6,2] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,5,2,1,6,4] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,3,6,4,5,2] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,2,4,1,5] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,2,6,3,1,5] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,3,4,2,6,5] => 0
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,3,2,4,6,7,5] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,1,5,6,3] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,1,6,4] => 0
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,3,6,2,5,7,4] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,2,3,4,1,5] => 0
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,3,7,4,2,6,5] => 0
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,3,6,7,4,5,2] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,3,6,4,2,7,5] => 0
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,3,4,7,2,6,5] => 0
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,3,7,4,5,6,2] => 0
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,3,4,2,6,7,5] => 0
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,3,6,4,5,7,2] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,3,4,7,5,6,2] => 0
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,3,4,5,2,7,6] => 0
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[] => [] => [1] => [1] => 0

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

$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3$

$F_{4} = 5$

Description

The number of nested exceedences of a permutation.
For a permutation

$\pi$ , this is the number of pairs

$i,j$ such that

$i < j < \pi(j) < \pi(i)$ . For exceedences, see St000155The number of exceedances (also excedences) of a permutation..

Map

Lehmer code rotation

Description

Sends a permutation

$\pi$ to the unique permutation

$\tau$ (of the same length) such that every entry in the Lehmer code of

$\tau$ is cyclically one larger than the Lehmer code of

$\pi$ .

Map

Ringel

Description

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

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.