Identifier
Values
[1] => [1,0,1,0] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 0
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 0
[2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 2
[] => [] => [] => [1,0] => 0
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Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.