- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000483: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 1

[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => 2

[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,5,8,7,6] => [2,3,4,5,6,1,8,7] => 3

[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,8,7,6,5,4,3,2] => [2,1,8,7,6,5,4,3] => 2

[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 2

[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,4,3,5,8,7,6] => [3,2,5,4,6,1,8,7] => 6

[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,6,5,7,8] => [3,2,5,4,7,6,8,1] => 6

[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => 5

[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,4,3,8,7,6,5] => [3,2,5,4,1,8,7,6] => 4

[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => [2,1,5,4,3,7,6,8] => [3,2,6,5,4,8,7,1] => 4

[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => 3

[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,8,7,6,5,4,3] => [3,2,1,8,7,6,5,4] => 2

[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [3,2,1,4,5,6,7,8] => [4,3,2,5,6,7,8,1] => 2

[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [3,2,1,8,7,6,5,4] => [4,3,2,1,8,7,6,5] => 2

[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => 3

[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [4,3,2,1,8,7,6,5] => [5,4,3,2,1,8,7,6] => 2

[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,8,7] => 2

[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [6,5,4,3,2,1,7,8] => [7,6,5,4,3,2,8,1] => 2

[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => 1

[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => 0

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => 1

[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,1] => 1

[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,9,8] => [2,3,4,5,6,7,8,1,9] => 2

[2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7,8,9] => [3,2,4,5,6,7,8,9,1] => 2

[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [7,6,5,4,3,2,1,9,8] => [8,7,6,5,4,3,2,1,9] => 1

[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8,7,6,5,4,3,2,1,9] => [9,8,7,6,5,4,3,2,1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 1 + 3\ q$

$F_{4} = 1 + 4\ q + 3\ q^{2}$

$F_{5} = 1 + 5\ q + 7\ q^{2} + 3\ q^{3}$

$F_{6} = 1 + 6\ q + 12\ q^{2} + 10\ q^{3} + 3\ q^{4}$

Description

The number of times a permutation switches from increasing to decreasing or decreasing to increasing.
This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

Kreweras complement

Description

Sends the permutation

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

$\pi^{-1}c$ where

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