Title: P-partitions, Dimer Models, and Higher Continued Fractions
Abstract: The continued fraction representations of rational numbers are related to certain products of SL(2,Z) matrices. Entries of these matrices can be seen to count several combinatorial objects, including lattice paths, perfect matchings of certain graphs, and order ideals of "fence posets". In recent work with Musiker, Schiffler, and Zhang, we studied the enumeration of "higher" versions of all these combinatorial objects (n-lattice paths, n-dimer covers, and P-partitions), using a product of SL(n,Z) matrices. I will discuss the background on these enumeration problems, as well as the notion of "higher continued fractions" which generalizes the correspondence in the SL(2,Z) case. Time permitting, I will also discuss work in progress on the q-analogs of these results.
