On the enumeration of double cosets and self-inverse double cosets

Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form $H\backslash G / H$ have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the informations on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.