Is it possible to shuffle a 2D matrix while preserving row AND column frequencies?

Question

Suppose I have a 2D array like the following:

GACTG
AGATA
TCCGA

Each array element is taken from a small finite set (in my case, DNA nucleo

Answer 1

It turns out that for 0-1 matrices, 2x2 swaps are sufficient to get from one matrix to any other. This was proved by H J Ryser as Theorem 3.1 in a paper called "Combinatorial Properties of Matrices of Zeros and Ones": http://cms.math.ca/cjm/v9/cjm1957v09.0371-0377.pdf . People have been trying to prove for a while that the Markov chain based on 2x2 swaps mixes rapidly; this paper http://arxiv.org/pdf/1004.2612v3 seems to come the closest.

If one could prove the generalization of Ryser's theorem to your case (maybe with up to 4x4 "swaps"), then on account of the symmetry of the swaps, it wouldn't be too hard to get a chain whose steady state distribution is uniform on the matrices of interest. I don't think there's any hope at the moment of proving that it mixes rapidly for all possible row/column distributions, but perhaps you know something about the distributions that we don't...