The matching problem in combinatorics asks: What is the probability of observing $k$ fixed points between an integer sequence and its random permutation? We can also express the question in terms of coding theory (What is the probability that a pair of randomly-generated strings with equal $n$ have a Hamming distance of $n - k$?) or estimation theory (If we draw n random pairs from a bivariate uniform population with correlation $r = 0$, then after converting each variable to ranks, what is the probability of there being k pairs with matching ranks?) The solution is well known: with increasing n, the distribution of k converges to $Poisson(1)$.
I'm interested in the answer to a specific generalization of the matching problem. I find it's easiest to express in the statistical framework: If we draw n random pairs from a bivariate population with correlation $r = \rho$, then after converting each variable to ranks, what is the probability of there being k pairs with matching ranks? In other words, what if the permutation is not random?
A lot has been written about the classical matching problem, and a lot about generalizations. For example, Diaconis and Holmes show that it makes very little difference if the bivariate distribution from which pairs are drawn is not uniform, but they retain the assumption of zero correlation. I've not been able to find this particular generalization. Any ideas?