- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 195 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

$F_{2} = 2$

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

$F_{4} = 4 + 10\ q$

$F_{5} = 9 + 33\ q$

$F_{6} = 22 + 104\ q + 6\ q^{2}$

Description

The number of cycles of length at least 3 of a permutation.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

Map

Ringel

Description

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

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].