Permutations as minimal powers of a single-cycle class-sum

The structure-constant [(p)]n · [(1)l1(2)l2 ⋯ (n)ln]n|[(1)l1−k1(2)l2 −k2⋯(u)lu+1⋯(n)ln]n that corresponds to the merging of the p cycles (1)k1(2)k2 ⋯ (u − 1)ku−1, (Σi=1u−1 ki = p) into a single cycle of length u = Σi=1u−1 ikl is derived. The result is used to evaluate the structure constant [(p)]nk|[(1)l1(p)l2⋯((i−1)(p−1)+1)li⋯]n, where k is the minimal power of [(P)]n generating the class-sum [(1)l1(p)l2⋯((i − 1)(p − 1) + 1)li⋯]n (i.e. Σi=1n (i − 1)li = k).