NonProjectiveInjectiveToNonInjectiveProjectiveHomDimension on DyckPath

The dimension of the homomorphisms from the direct sum of the non-projective indecomposable injective modules to the direct sum of the non-injective indecomposable projective modules.

On DyckPath

FindStat's St000687. A projective is injective when the projective before it is no longer than it.

Called as CombinatorialStat(x, "NonProjectiveInjectiveToNonInjectiveProjectiveHomDimension"). The head's page has its examples.

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FindStat St000687

The dimension of $\operatorname{Hom}(I,P)$ for the linear Nakayama algebra corresponding to a Dyck path.

The dimension of $\operatorname{Hom}(I,P)$ for the linear Nakayama algebra corresponding to a Dyck path. Here $I$ is the direct sum of all indecomposable injective non-projective modules and $P$ is the direct sum of all indecomposable projective non-injective modules. The correspondence between linear Nakayama algebras and Dyck paths is explained on the [[https://www.findstat.org/CollectionsDatab…

Object	Value
[1,0]	0
[1,0,1,0]	0
[1,1,0,0]	0
[1,0,1,0,1,0]	0
[1,0,1,1,0,0]	0
[1,1,0,0,1,0]	0
[1,1,0,1,0,0]	0
[1,1,1,0,0,0]	0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000687 has the rest.