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ThreeDimensionalModulesOfProjectiveDimensionOne on DyckPath ​

The indecomposable modules of dimension three and projective dimension one.

On DyckPath

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

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FindStat St001234

The number of indecomposable three dimensional modules with projective dimension one.

The number of indecomposable three dimensional modules with projective dimension one. It return zero when there are no such modules.
ObjectValue
[1,0]0
[1,0,1,0]0
[1,1,0,0]0
[1,0,1,0,1,0]0
[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]1

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001234 has the rest.