In their work on `Coxeter-like complexes', Babson and Reiner introduced a
simplicial complex ΔT associated to each tree T on n nodes,
generalizing chessboard complexes and type A Coxeter complexes. They
conjectured that ΔT is (n−b−1)-connected when the tree has b
leaves. We provide a shelling for the (n−b)-skeleton of ΔT, thereby
proving this conjecture.
In the process, we introduce notions of weak order and inversion functions on
the labellings of a tree T which imply shellability of ΔT, and we
construct such inversion functions for a large enough class of trees to deduce
the aforementioned conjecture and also recover the shellability of chessboard
complexes Mm,n with n≥2m−1. We also prove that the existence or
nonexistence of an inversion function for a fixed tree governs which networks
with a tree structure admit greedy sorting algorithms by inversion elimination
and provide an inversion function for trees where each vertex has capacity at
least its degree minus one.Comment: 23 page