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ModulesOfProjectiveDimensionAtMostOne on DyckPath ​

The indecomposable modules of projective dimension at most one.

On DyckPath

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

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FindStat St001003

The number of indecomposable modules with projective dimension at most one in the linear Nakayama algebra corresponding to a Dyck path.

ObjectValue
[1,0]3
[1,0,1,0]4
[1,1,0,0]6
[1,0,1,0,1,0]5
[1,0,1,1,0,0]6
[1,1,0,0,1,0]7
[1,1,0,1,0,0]7
[1,1,1,0,0,0]10

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001003 has the rest.