Extend the word-encoding of facets and shelling to larger minors

Develop a formulation that encodes facets of the simplicial complex Δ associated to the initial ideals of double determinantal ideals I_{mn}^r generated by t-minors with t>2 as words on an appropriate alphabet, and ascertain whether an analogous shelling order exists based on adjacent transpositions of these words.

Background

For 2-minors, the paper provides a combinatorial description of facets of Δ as unions of r paths and a bijection to words on the multiset {M^{m−1}, N^{n−1}, R^{r−1}} (Proposition NMR).

The authors then pose whether this word-based facet encoding and the associated shelling considerations can be generalized when the generating minors have size larger than 2.