Séminaire Lotharingien de Combinatoire, 84B.26 (2020), 12 pp.

Sergi Elizalde

Measuring Symmetry in Lattice Paths and Partitions

Abstract. We introduce the notion of degree of symmetry for lattice paths and related combinatorial objects. The degree of symmetry measures how symmetric an object is, usually ranging from zero (completely asymmetric) to its size (completely symmetric). We study the behavior of this statistic on Dyck paths and grand Dyck paths, where the symmetry is given by reflection along a vertical line through their midpoint; partitions, where the symmetry is 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 generating functions are algebraic in most cases, with the notable exception of Dyck paths, for which we apply techniques from walks in the plane to find a functional equation for the generating function, and conjecture that it is D-finite but not algebraic.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

The following versions are available: