ProjectiveRadicalsOfVanishingFirstExtensionFromJacobsonRadical on DyckPath
The indecomposable projective modules whose radical, zero included, has no first extension by the Jacobson radical.
FindStat's St001226: the number of i with Ext^1(J, e_i J) zero. The last projective is simple, so its radical is zero and counts.
Called as CombinatorialStat(x, "ProjectiveRadicalsOfVanishingFirstExtensionFromJacobsonRadical"). The head's page has its examples.
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FindStat St001226
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
| Object | Value |
|---|---|
[1,0] | 2 |
[1,0,1,0] | 2 |
[1,1,0,0] | 3 |
[1,0,1,0,1,0] | 2 |
[1,0,1,1,0,0] | 3 |
[1,1,0,0,1,0] | 3 |
[1,1,0,1,0,0] | 2 |
[1,1,1,0,0,0] | 4 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001226 has the rest.