Shelling orders via adjacent transpositions of facet words for Δ

Determine whether, for the simplicial complex Δ that is the Stanley–Reisner complex of the initial ideal of the toric double determinantal ideal I_{mn}^r generated by 2-minors (with respect to any diagonal term order), any ordering of the facets in which consecutive facets correspond to words on the multiset {M^{m−1}, N^{n−1}, R^{r−1}} that differ by exchanging a single adjacent pair is a shelling order of Δ.

Background

Section 5 defines Δ as the simplicial complex associated to the initial ideal of I_{mn}^r (2-minors) with respect to a diagonal term order, and characterizes its facets as unions of paths.

Proposition NMR establishes a bijection between facets of Δ and words on the multiset {M^{m−1}, N^{n−1}, R^{r−1}} by encoding steps within and transitions between the r path-components.

In the concluding discussion, the authors note an example where ordering facets so that successive facets correspond to words differing by swapping an adjacent pair yields a shelling; they then ask whether any such adjacent-transposition-based ordering always gives a shelling order.