- FindStat

Identifier

Mp00071: Permutations —descent 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] => [1,1] => [1,0,1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

bounce path

Description

The bounce path determined by an integer composition.

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.