- FindStat

Identifier

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 624 entries. <<<

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$F_{2} = 2\ q$

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

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

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

$F_{6} = 6\ q + 40\ q^{2} + 60\ q^{3} + 24\ q^{4} + 2\ q^{5}$

$F_{7} = 7\ q + 70\ q^{2} + 175\ q^{3} + 140\ q^{4} + 35\ q^{5} + 2\ q^{6}$

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

Map

switch returns and last double rise

Description

An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.