The conjugacy classes of the permutation group $S_n$ are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$ describing the cycle type. What happens when you take products of two whole conjugacy classes? I saw in a paper, $$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$ Which I take to mean if you multiply a 6-cycle (abcdef) and a product of disjoint 3-cycles (pq)(rs)(tv), you can get

• a three-cycle (abc),
• two two-cycles (ab)(cd),
• a five-cycles (abcde),
• a four-cycles and a two-cycle (abcd)(ef),
• two three cycles (abc)(def)

With certain multiplicities. Is it predictable what kinds of conjugacy classes you get? Is there an interpretation of this as the intersection cohomology of some moduli space?