  Dlab and Ringel showed that algebras being quasi-hereditary in all 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, and if the algebra with
permuted indices is quasi-hereditary, then we say that this permutation gives a
quasi-hereditary structure. In this article, we first give a criterion for
adjacent transpositions giving quasi-hereditary structures, in terms of
homological conditions of standard or costandard modules over a given
quasi-hereditary algebra. Next, we consider those which we call connectedness
of quasihereditary structures. The definition of connectedness can be found in
Definition 4.1. We then show that any two quasi-hereditary structures are
connected, which is our main result. By this result, once we know that there
are two quasi-hereditary structures, then permutations in some sense lying
between them give also quasi-hereditary structures.
