- FindStat

Identifier

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

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

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

$F_{5} = 16 + 12\ q + 21\ q^{2} + 22\ q^{3} + 23\ q^{4} + 12\ q^{5} + 11\ q^{6} + 2\ q^{7} + q^{8}$

$F_{6} = 32 + 32\ q + 62\ q^{2} + 80\ q^{3} + 106\ q^{4} + 102\ q^{5} + 102\ q^{6} + 78\ q^{7} + 62\ q^{8} + 36\ q^{9} + 18\ q^{10} + 8\ q^{11} + 2\ q^{12}$

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

DEX composition

Description

The DEX composition of a permutation.
Let

$\pi$ be a permutation in

$\mathfrak S_n$ . Let

$\bar\pi$ be the word in the ordered set

$\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from

$\pi$ by replacing every excedance

$\pi(i) > i$ by

$\overline{\pi(i)}$ . Then the DEX set of

$\pi$ is the set of indices

$1 \leq i < n$ such that

$\bar\pi(i) > \bar\pi(i+1)$ . Finally, the DEX composition

$c_1, \dots, c_k$ of

$n$ corresponds to the DEX subset

$\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$ .
The (quasi)symmetric function

$\sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)},$
where the sum is over the set of permutations of cycle type

$\lambda$ with

$j$ excedances, is the Eulerian quasisymmetric function.

Map

bounce path

Description

The bounce path determined by an integer composition.