- FindStat

Identifier

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001003: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 3
[1,0,1,0] => [1,0,1,0] => 4
[1,1,0,0] => [1,1,0,0] => 6
[1,0,1,0,1,0] => [1,0,1,0,1,0] => 5
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 7
[1,1,0,0,1,0] => [1,1,0,0,1,0] => 7
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 6
[1,1,1,0,0,0] => [1,1,1,0,0,0] => 10
[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] => [1,1,0,1,0,1,0,0] => 8
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 8
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 7
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 11
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 8
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 11
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 7
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 7
[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 9
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 11
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 9
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 9
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 15
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 9
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 9
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 8
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 12
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => 9
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 12
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => 8
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 8
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 10
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 12
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 10
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 10
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 16
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => 12
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 12
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 10
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 16
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 8
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 10
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 10
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 10
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 10
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 9
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 12
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 12
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 16
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 10
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 10
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 13
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 10
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 13
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 16
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 13
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 12
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 13
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 21

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$F_{1} = q^{3}$

$F_{2} = q^{4} + q^{6}$

$F_{3} = q^{5} + q^{6} + 2\ q^{7} + q^{10}$

$F_{4} = q^{6} + 3\ q^{7} + 3\ q^{8} + 3\ q^{9} + 3\ q^{11} + q^{15}$

$F_{5} = q^{7} + 6\ q^{8} + 6\ q^{9} + 12\ q^{10} + 8\ q^{12} + 4\ q^{13} + 4\ q^{16} + q^{21}$

Description

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

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.