A recursion of the Pólya polynomial for the symmetric group

A recursion for the Pólya polynomial of the symmetric group is proved. This recursion will be used to compute the Pólya polynomial of the Young group. In turn, a theorem by George Pólya, generalized by deBruijn, will be used to compute the double cosets of Young groups embedded in an overall symmetric group. Connections to mathematical physics and combinatorics will be noted but not proved.