Symmetries of statistics on lattice paths between two boundaries

TL;DR

This paper proves symmetry in the joint distribution of certain path statistics between two boundaries, using involutions and matroid properties, with applications to various combinatorial structures.

Contribution

It introduces a new involution-based method to establish symmetry of path statistics and extends results to multiple paths and related combinatorial objects.

Findings

Symmetry in the joint distribution of shared E steps with boundaries.

Extension of symmetry results to k-tuples of paths and applications.

A new bijection linking path tuples to Young tableaux.

Abstract

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. Finally, we prove a conjecture of Nicol\'as 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.