## Counting cycles after permuting within rows and columns

Consider a rectangular p×q array, labelled by the numbers 0,…,pq−1 for convenience. Let Sp and Sq and Spq denote the symmetric groups. Take a family of permutations: ρ1,…,ρq∈Sp, κ1,…,κp∈Sq. Then letting ρt act on the tth row, one obtains a permutation ρ∈Spq which leaves the rows of the array stable (but permutes within each row). … Read more