Towards combinatorial derivations of K-polynomials for determinantal varieties

Let ${\mathfrak{X}}_k\subseteq {\mathsf{M}\mathsf{a}\mathsf{t}}_{m,n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of ${\mathfrak{X}}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0<k<\min \{m,n\}$. When $k=1$ or $k=\min \{m,n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood–Richardson tableaux.