- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 6 + 6\ q + 4\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{5} = 24 + 24\ q + 18\ q^{2} + 18\ q^{3} + 14\ q^{4} + 8\ q^{5} + 6\ q^{6} + 4\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 120 + 120\ q + 96\ q^{2} + 96\ q^{3} + 84\ q^{4} + 66\ q^{5} + 46\ q^{6} + 30\ q^{7} + 22\ q^{8} + 16\ q^{9} + 10\ q^{10} + 6\ q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

Description

The bounce statistic of a Dyck path.
The bounce path

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ .
The points where

$D'$ touches the

$x$ -axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ .
In particular, the bounce statistics of

$D$ and

$D'$ coincide.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.