  We study the commutative algebras $Z_{JK}$ appearing in Brown and Goodearl's
extension of the $\mathcal{H}$-stratification framework, and show that if $A$
is the single parameter quantized coordinate ring of $M_{m,n}$, $GL_n$ or
$SL_n$, then the algebras $Z_{JK}$ can always be constructed in terms of
centres of localizations. The main purpose of the $Z_{JK}$ is to study the
structure of the topological space $spec(A)$, which remains unknown for all but
a few low-dimensional examples. We explicitly construct the required
denominator sets using two different techniques (restricted permutations and
Grassmann necklaces) and show that we obtain the same sets in both cases. As a
corollary, we obtain a simple formula for the Grassmann necklace associated to
a cell of totally nonnegative real $m\times n$ matrices in terms of its
restricted permutation.
