Count Ringel-self-dual orientations of A_{4m+1}

Determine how many distinct orientations of the Dynkin diagram A_{4m+1} yield a path algebra admitting a Ringel-self-dual quasi-hereditary structure.

Background

The paper proves that type A_{4m+1} is among the Dynkin types for which some orientation and partial ordering produce Ringel self-duality. It does not determine the number of orientations with this property, so the enumeration of such orientations remains unresolved.