A MINIMAL-RANK COUNTEREXAMPLE TO SHELLABILITY IN THE QUASISYMMETRIC BRUHAT ORDER

We give a counterexample to Bansal’s shellability conjecture for the quasisymmetric Bruhat order on indexed binary forests. For the forests with codes E = (0, 1, 0, 1, 1, 0, 1) and F = (1, 2, 4), the order complex of (E, F) is the disjoint union of two six-cycles. This complex is a face link in the order complex of (∅, F), so the principal lower interval [∅, F] is not shellable. We verify the cover table by exact forest-polynomial coefficient extraction and show, using recursive coatom orderings in rank six, that seven is the smallest possible rank of a counterexample top. A standalone verifier and the finite coefficient certificate accompany the paper.