- FindStat

Identifier

Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001523: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The degree of symmetry of a Dyck path.
Given a Dyck path $D=(d_1,\dots,d_{2n})$, with $d_i\in\{0,1\}$, this is the number of positions $1\leq i\leq n$ such that $d_i = 1-d_{2n+1-i}$ and the initial height of the $i$-th step $\sum_{j < i} d_i$ equals the final height of the $(2n+1-i)$-th step.

Map

peeling map

Description

Send a Dyck path to its peeled Dyck path.