A remark of the number of quasi-hereditary structures

Dlab and Ringel showed that algebras being quasi-hereditary in all total orders for indices of primitive idempotents becomes hereditary. So, we are interested in for which orders a given quasi-hereditary algebra is again quasi-hereditary. As a matter of fact, we consider permutations of indices instead of total orders. If the standard modules defined by two permutations coincide, we say that the permutations are equivalent. Moreover if the algebra with permuted indices is quasi-hereditary, then this equivalence class of the permutation is called a quasi-hereditary structure. In this article, we give a method of counting the number of quasi-hereditary structures for certain algebras.