- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000541: Permutations ⟶ ℤ (values match St000542The number of left-to-right-minima of a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 234 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

$F_{2} = 1 + q$

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

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

$F_{5} = 9 + 16\ q + 12\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 22 + 44\ q + 40\ q^{2} + 20\ q^{3} + 5\ q^{4} + q^{5}$

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation

$\pi$ of length

$n$ , this is the number of indices

$2 \leq j \leq n$ such that for all

$1 \leq i < j$ , the pair

$(i,j)$ is an inversion of

$\pi$ .

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

to 132-avoiding permutation

Description

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