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Triangle-free triangulations, hyperplane arrangements and shifted tableaux. (English) Zbl 1253.05032

Summary: 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 a pair of distinguished antipodes;
(2)
the number of geodesics between these antipodes is equal to twice the number of standard Young tableaux of a truncated shifted staircase shape.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05E18 Group actions on combinatorial structures
05E10 Combinatorial aspects of representation theory
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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