List of Publications with Abstracts

We prove that on the set of lattice paths with steps N = (0, 1) and E = (1, 0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics ‘number of E steps shared with B’ and ‘number of E steps shared with T ’ have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S = (0,−1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, to pattern-avoiding permutations, and to the generalized Tamari lattice.