Given two $k$-subsets $I$ and $J$ of $\{1 \dots n\}$, denote by $\min(J)$ the minimal element in $J$ and by $\max(I)$ the maximal element in $I$, we write $I \prec J$ if $\max(I)<\min(J)$. The sets $I$ and $J$ are called weakly separated if at least one of the following two conditions holds:

(1) $J-I$ can be partitioned into a disjoint union $J-I = J' \sqcup J''$ so that $J' \prec I-J \prec J''$;

(2) $I-J$ can be partitioned into a disjoint union $I-J = I' \sqcup I''$ so that $I' \prec J-I \prec I''$.

This condition is introduced in Leclerc and Zelevinsky's paper Quasi-commuting families of quantum minors.

Are there some method in combinatorics to count the number of all weakly separated pairs $I, J$ where $|I|=|J|=k$ and $I, J \subset \{1, \ldots, n\}$? Thank you very much.