- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 6

[1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 5

[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 5

[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 6

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

[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 5

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

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

[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 5

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 8

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 7

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 7

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 9

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 6

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 8

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 6

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 8

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 7

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 7

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 7

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 8

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 7

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 6

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 6

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 7

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 5

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 6

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 5

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 7

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 6

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 6

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 6

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 7

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 5

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 6

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 5

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 5

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 6

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 5

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 5

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 5

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 6

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Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.