A character theoretic formula for the base size

A base for a permutation group <i>G</i> acting on a set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> is a sequence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> of points of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> such that the pointwise stabiliser <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:mi>B</mml:mi></mml:msub> </mml:math> is trivial. The base size of <i>G</i> is the size of a smallest base for <i>G</i>. 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 <i>G</i> of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>l</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>.</mml:mo></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>S</mml:mtext> <mml:mi>n</mml:mi></mml:msub> </mml:math> acting on the <i>k</i>-element subsets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> <mml:mo>.</mml:mo></mml:mrow> </mml:math> Our methods also provide a formula for the base size of many product type permutation groups.