- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001033: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with St001034The area of the parallelogram polyomino associated with the Dyck path. and with St000395The sum of the heights of the peaks of a Dyck path..

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 $1 \tilde D_2 0 \tilde D_1$.