- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000354: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 195 entries. <<<

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searching the database for the individual values of this statistic

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searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 2\ q$

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

$F_{4} = 6\ q^{2} + 8\ q^{3}$

$F_{5} = 2\ q^{2} + 24\ q^{3} + 16\ q^{4}$

$F_{6} = 20\ q^{3} + 80\ q^{4} + 32\ q^{5}$

Description

The number of recoils of a permutation.
A recoil, or inverse descent of a permutation

$\pi$ is a value

$i$ such that

$i+1$ appears to the left of

$i$ in

$\pi_1,\pi_2,\dots,\pi_n$ .
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern

$([2,1], {(0,1),(1,1),(2,1)})$ , i.e., the middle row is shaded.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].