SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule on DyckPath
The simple modules that are not in the socle of the regular module or have no first extension by the regular module.
FindStat's St001492. A simple module is in the socle when some indecomposable projective ends at it.
Called as CombinatorialStat(x, "SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule"). The head's page has its examples.
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FindStat St001492
The number of simple modules of the linear Nakayama algebra corresponding to a Dyck path that do not appear in the socle of the regular module, or have no nontrivial self-extensions with the regular module.
| Object | Value |
|---|---|
[1,0] | 2 |
[1,0,1,0] | 2 |
[1,1,0,0] | 3 |
[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] | 4 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001492 has the rest.