SimplesOfEvenProjectiveDimension on DyckPath

The simple modules of even projective dimension.

On DyckPath

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

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FindStat St001165

The number of simple modules with even projective dimension of the linear Nakayama algebra corresponding to a Dyck path.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001165 has the rest.

FindStat St001471

The magnitude of a Dyck path.

The magnitude of a Dyck path. The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C. We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001471 has the rest.