Séminaire Lotharingien de Combinatoire, 84B.26 (2020), 12 pp.
Measuring Symmetry in Lattice Paths and Partitions
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
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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