- FindStat

Identifier

Mp00028: Dyck paths —reverse⟶ Dyck paths
St001294: Dyck paths ⟶ ℤ

Values

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

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

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

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

[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] => 1

[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] => 3

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

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

[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] => 2

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

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

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

[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] => 1

[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] => 4

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

[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] => 2

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

[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] => 1

[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] => 3

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

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

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

[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] => 3

[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] => 1

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

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

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

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

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

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

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

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

[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] => 2

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

[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] => 2

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

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

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

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

[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] => 1

[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] => 5

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

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

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

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

[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] => 2

[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] => 4

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

[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] => 1

[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] => 3

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

>>> Load all 196 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2}$

$F_{4} = 1 + 8\ q + 4\ q^{2} + q^{3}$

$F_{5} = 1 + 20\ q + 15\ q^{2} + 5\ q^{3} + q^{4}$

$F_{6} = 1 + 50\ q + 52\ q^{2} + 22\ q^{3} + 6\ q^{4} + q^{5}$

Description

The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers oeis.org/A001006.

Map

reverse

Description

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