A character theoretic formula for the base size

A base for a permutation group G acting on a set $$ is a sequence $$ of points of $$ such that the pointwise stabiliser $$ is trivial. The base size of G is the size of a smallest base for G. We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for G of size $$ As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group $$ acting on the k-element subsets of $$ Our methods also provide a formula for the base size of many product type permutation groups.