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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.

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Acknowledgements

The author thanks Emeric Deutsch for suggesting the notion of degree of asymmetry of combinatorial objects, Alin Bostan for helpful discussions about D-finiteness, and two reviewers for insightful advice on improving the content and the presentation of this paper.

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Correspondence to Sergi Elizalde.

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Communicated by Frédérique Bassino.

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Elizalde, S. The Degree of Symmetry of Lattice Paths. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00551-6

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Keywords

  • Symmetry
  • Dyck path
  • Partition
  • Bargraph
  • Lattice walk

Mathematics Subject Classification

  • 05A15
  • 05A19
  • 05A17