Counting Double Cosets Using Maple Bill Pletsch Central New Mexico Community College Albuquerque, New Mexico, USA Unlike the coset, the double coset is poorly behaved. Counting the number of double cosets a subgroup generates in a group can be difficult. At ACA2007, it was shown how the number of Young double cosets in an overall symmetric group can be counted using the enumeration theory of George Polya. While doing so some interesting patterns were demonstrated, but not proved. This year these patterns will be proved. To do so first a canonical form is defined. Maple is induced to generate a transversal of the double cosets in the canonical form. Some set mappings are defined and used in Maple in a manner analogous to the method of subtraction used on ordinary data sets. Sets are subtracted instead of numbers. A final induction on the remaining sets finishes the proof.