4 research outputs found
Alexander Duality and Rational Associahedra
A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong,
Williams, and the author initiated the systematic study of {\em rational
Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics
(which is in turn a generalization of classical Catalan combinatorics). The
latter paper gave two possible models for a rational analog of the
associahedron which attach simplicial complexes to any pair of coprime positive
integers a < b. These complexes coincide up to the Fuss-Catalan level of
generality, but in general one may be a strict subcomplex of the other.
Verifying a conjecture of Armstrong, Williams, and the author, we prove that
these complexes agree up to homotopy and, in fact, that one complex collapses
onto the other. This reconciles the two competing models for rational
associahedra. As a corollary, we get that the involution (a < b)
\longleftrightarrow (b-a < b) on pairs of coprime positive integers manifests
itself topologically as Alexander duality of rational associahedra. This
collapsing and Alexander duality are new features of rational Catalan
combinatorics which are invisible at the Fuss-Catalan level of generality.Comment: 23 page
Recommended from our members
Enumerative Combinatorics
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds