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ThreeRegularSimpleModules on DyckPath ​

The simple modules that are 3-regular.

On DyckPath

FindStat's St001137.

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

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FindStat St001137

The number of 3-regular simple modules of the linear Nakayama algebra corresponding to a Dyck path.

ObjectValue
[1,0]0
[1,0,1,0]0
[1,1,0,0]0
[1,0,1,0,1,0]1
[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/St001137 has the rest.