We present a combinatorial method to determine the dimension of $C{H}$-strata in the algebra of $mtimes n$ quantum matrices $Oq$ as follows. To a given $C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $C{H}$-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 $C{H}$-strata in $Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $C{H}$-strata in $Oq$.
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