SimplesOfNonMaximalProjectiveDimension on DyckPath
The simple modules whose projective dimension is less than the global dimension.
FindStat's St001170 counts the indecomposable injective modules whose socle is such a simple module: there is one injective for each simple socle.
Called as CombinatorialStat(x, "SimplesOfNonMaximalProjectiveDimension"). The head's page has its examples.
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FindStat St001170
The number of indecomposable injective modules whose socle has projective dimension at most the global dimension of the algebra minus one of the linear Nakayama algebra corresponding to a Dyck path.
| Object | Value |
|---|---|
[1,0] | 1 |
[1,0,1,0] | 2 |
[1,1,0,0] | 1 |
[1,0,1,0,1,0] | 3 |
[1,0,1,1,0,0] | 3 |
[1,1,0,0,1,0] | 3 |
[1,1,0,1,0,0] | 3 |
[1,1,1,0,0,0] | 1 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001170 has the rest.