- FindStat

Identifier

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000662: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

$F_{5} = 3\ q + 16\ q^{2} + 20\ q^{3} + 3\ q^{4}$

Description

The staircase size of the code of a permutation.
The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ .
The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ .
This statistic is mapped through Mp00062Lehmer-code to major-code bijection to the number of descents, showing that together with the number of inversions St000018The number of inversions of a permutation. it is Euler-Mahonian.

Map

peaks-to-valleys

Description

Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus

$2$ .
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.

Map

Ringel

Description

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

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.