The degree of symmetry of lattice paths

Sergi Elizalde

Published 2020-02-28Version 1

The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry, and their semilength or semiperimeter. The resulting generating functions are algebraic in most cases, with the notable exception of Dyck paths, for which we apply bijections to walks in the plane in order to find a functional equation for the corresponding generating function, which we conjecture to be D-finite but not algebraic.