SimplesOfGradeAtLeastTwo on DyckPath

The simple modules of grade at least two.

On DyckPath

FindStat's St001188, and St001244 (the simples of projective dimension one that are not 1-regular), which has the same values on every path of semilength 8 or less.

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

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Sizes 1 to

FindStat St001188

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.

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]	1
[1,1,1,0,0,0]	0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001188 has the rest.

FindStat St001244

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.

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]	1
[1,1,1,0,0,0]	0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001244 has the rest.