Lagrange Collisions and Cover Relations for Rational Dyck Paths

Abstract

Schiffler associated matching and Lagrange orders to rational Dyck paths and posed two problems about their equality and cover relations. We first classify the band-graph isomorphism classes at a fixed coprime endpoint: every class is an orbit of an explicit reversal involution and therefore has size at most two. We then give two paths in D(17,9) with the same exact Lagrange value but in different band-graph orbits, answering the equality question negatively. Two independent exact enumerations through total length 25 show that the length-26 example is computationally minimal. For the cover problem, we attach exact matrix intervals to rational-Dyck prefixes. A best-first traversal emits complete equality levels in increasing order for both scores; equivalently, a pair is a cover precisely when a finite prefix antichain certifies all path cylinders outside its open score interval. We supplement this global characterization with structural matching results: the parity of every matching score is determined by the endpoint, every score gap of two is a cover, a local interchange has an exact four-case continuant formula, and the first n matching levels in D(n,n-1) are explicit singletons. The last result yields covers at Hamming distance 2n-6, ruling out any uniformly bounded local-move catalogue. All finite certificates use integer arithmetic or reduced rational squares.