Discoveries / combinatorics

The Hamaker-Reiner Conjecture on ASM Weak Order Intervals

Hamaker and Reiner conjectured that the order complex of an open interval $(u,w)$ in the ASM weak order is contractible unless $w$ is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on $S_n$ whose order complex is not contractible even though $w$ has no such form, detected by a nonzero Mobius function value.