Enumeration of H-strata in quantum matrices with respect to dimension

We present a combinatorial method to determine the dimension of H-strata in the algebra of m×n quantum matrices Oq(Mm,n(K)) as follows. To a given H-stratum we associate a certain permutation via the notion of pipe dreams. We show that the dimension of the H-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the d-dimensional H-strata in Oq(Mm,n(K)). Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors (Bell and Launois (2010) [3], , Bell, Launois and Lutley (2010) [4]) regarding the asymptotic proportion of d-dimensional H-strata in Oq(Mm,n(K)).