In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the '90s, proofs of integrality were known only for individual special cases. In the early '00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson's integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually. | T tT-1 To s en Ekedahl Stockhohns uuivcrsitct SE-Í06 91 Stockholm . . Submitted Oct 31 2008 Accepted Dec 7 2009 Published Dec 13 2009 . . Abstract. H e lutroduco a sequence of poset close y related to he . In this wc are motivated by reasons similar to those of Stasheff in the case of the associahedra. We make a study of this poset showing that it has an inductive structure with proper downwa ds lut reals being prods ts o smaller pose s in e same series and associahcdra. u ing this wc also show that they are hi t dual CL-shdlab e and 1 . . The purpose of tins a ti le is to introduce a sequence of posers closely related to the face lattices of the associahedra and study their combinatorial properties in particular it will be shown that they are shellable. The origin of these posets are in principle not relevant for such a study nevertheless I shall start by bnefly discussing it The aasociahedra are relevant o the description of products winch are assoaatrve only up o homotopy A. -spaces . The prototyp cal such example IS the path space of a topological span where the composition of paths is not associative but is associative up to homotopy and two maps from one space to another constructed out of such homotopms are homotopic and so on. Suppos now that he IS a manifold M and that we are really only interested in smooth paths. The problem is that the composition of smooth paths is usually not smooth. The solution would seem to be to smooth the composition but the problem then is that such a smoothing is not unique. Thus one is forced to speak about a composition and will have to contend with the ambiguities inherent in that. A direct way of expressing a composition is as a smooth map from the standard 2-simplex A2 to THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R23 1 M Where the origins two paths are the restriction of tha map to the first two edges and the par ieular compos to s the restriction to the thud edge. Th map tse
