  We study standard monomial bases for Richardson varieties inside the flag
variety. In general, writing down a standard monomial basis for a Richardson
variety can be challenging, as it involves computing so-called defining chains
or key tableaux. However, for a certain family of Richardson varieties, indexed
by compatible permutations, we provide a very direct and straightforward
combinatorial rule for writing down a standard monomial basis. We apply this
result to the study of toric degenerations of Richardson varieties. In
particular, we provide a new family of toric degenerations of Richardson
varieties inside flag varieties.
