The degree of symmetry of lattice paths
Abstract
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, deducing in most cases that, asymptotically, the degree of symmetry has a Rayleigh or half-normal limiting distribution. The resulting generating functions are often algebraic, with the notable exception of Dyck paths, for which we conjecture that it is D-finite (but not algebraic), based on a functional equation that we obtain using bijections to walks in the plane.
Keywords
Cite
@article{arxiv.2002.12874,
title = {The degree of symmetry of lattice paths},
author = {Sergi Elizalde},
journal= {arXiv preprint arXiv:2002.12874},
year = {2021}
}
Comments
31 pages, 10 figures, 4 tables. The main additions to this version are the analysis of the limiting distributions of the degree of symmetry in the various cases, a new subsection 5.2 relating certain bargraphs to peakless Motzkin paths, minor corrections and improvements, and additional references