- FindStat

Identifier

Mp00028: Dyck paths —reverse⟶ Dyck paths
St001036: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + 2\ q$

$F_{4} = 4 + 8\ q + 2\ q^{2}$

$F_{5} = 5 + 20\ q + 15\ q^{2} + 2\ q^{3}$

$F_{6} = 6 + 40\ q + 60\ q^{2} + 24\ q^{3} + 2\ q^{4}$

$F_{7} = 7 + 70\ q + 175\ q^{2} + 140\ q^{3} + 35\ q^{4} + 2\ q^{5}$

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Map

reverse

Description

The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.