SimpleModulesSecondExtensionWithRegularModuleDimension on DyckPath

The dimension of the second extensions of the direct sum of the simple modules by the regular module.

On DyckPath

FindStat's St001215.

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

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FindStat St001215

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001215 has the rest.