Face rings of cycles, associahedra, and standard Young tableaux
Abstract
We show that Jn, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron An. This resolution is not minimal for n > 5; in this case the Betti numbers of Jn are strictly smaller than the f-vector of An. We show that in fact the Betti numbers of Jn are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1n-d-3). This complements the fact that the number of (d-1)-dimensional faces of An are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1n-d-3); a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of Jn that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.