SocleSimplesOfVanishingFirstExtensionWithRegularModule on DyckPath

The simple modules in the socle of the regular module that have no first extension by the regular module.

On DyckPath

FindStat's St001483. A simple module is in the socle when some indecomposable projective ends at it.

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

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Distribution

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Sizes 1 to

FindStat St001483

The number of simple modules $S$ that occur in the socle of the regular module $A$ of the linear Nakayama algebra corresponding to a Dyck path and satisfy $\operatorname{Ext}^1_A(S,A)=0$.

Object	Value
[1,0]	1
[1,0,1,0]	1
[1,1,0,0]	1
[1,0,1,0,1,0]	2
[1,0,1,1,0,0]	1
[1,1,0,0,1,0]	1
[1,1,0,1,0,0]	1
[1,1,1,0,0,0]	1

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001483 has the rest.