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TwoDimensionalModulesOfProjectiveDimensionOne on DyckPath ​

The indecomposable modules of dimension two and projective dimension one.

On DyckPath

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

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FindStat St001233

The number of indecomposable 2-dimensional modules with projective dimension one.

ObjectValue
[1,0]0
[1,0,1,0]0
[1,1,0,0]1
[1,0,1,0,1,0]0
[1,0,1,1,0,0]0
[1,1,0,0,1,0]1
[1,1,0,1,0,0]1
[1,1,1,0,0,0]2

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001233 has the rest.