Abstract
A base for a permutation group G acting on a set Ω is a sequence B of points of Ω such that the pointwise stabiliser GB 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 l ∈ N. 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 Sn acting on the k-element subsets of {1, 2, 3, …, n}. Our methods also provide a formula for the base size of many product type permutation groups.