Symmetries of statistics on lattice paths between two boundaries

We prove that on the set of lattice paths with steps N=(0,1)N=(0,1) and E=(1,0)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)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.Finally, we prove a conjecture of Nicolás about the distribution of degrees of k consecutive vertices in k-triangulations of a convex n-gon. To achieve this goal, we provide a new statistic-preserving bijection between certain k-tuples of non-crossing paths and k-flagged semistandard Young tableaux, which is based on local moves reminiscent of jeu de taquin.