  Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.

Combinatorics