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Combinatorics of periodic ellipsoidal billiards

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Abstract

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.

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Acknowledgements

The authors are grateful to the referee for suggestions that improved the presentation.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, State College, USA

    George E. Andrews

  2. Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, USA

    Vladimir Dragović

  3. School of Mathematics and Statistics, The University of Sydney, Camperdown, Australia

    Milena Radnović

  4. Mathematical Institute SANU, Belgrade, Serbia

    Vladimir Dragović & Milena Radnović

Authors
  1. George E. Andrews
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  2. Vladimir Dragović
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  3. Milena Radnović
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Correspondence to Vladimir Dragović.

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In honor of Richard Askey, a great man and a great mathematician

GEA was partially supported by Simons Foundation Grant #633284. The research of VD and MR was supported by the Discovery Project #DP190101838 Billiards within confocal quadrics and beyond from the Australian Research Council, the Serbian Ministry of Education, Technological Development and Science and the Science Fund of Serbia. VD would like to thank the University of Sydney Mathematical Research Institute and their International Visitor Program for kind hospitality.

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Andrews, G.E., Dragović, V. & Radnović, M. Combinatorics of periodic ellipsoidal billiards. Ramanujan J 61, 135–147 (2023). https://doi.org/10.1007/s11139-020-00346-y

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  • DOI: https://doi.org/10.1007/s11139-020-00346-y

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