- FindStat

Identifier

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

Map

reflect parallelogram polyomino

Description

Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.