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Sweeping up zeta

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Abstract

We repurpose the main theorem of Thomas and Williams (J Algebr Comb 39(2):225–246, 2014) to prove that modular sweep maps are bijective. We construct the inverse of the modular sweep map by passing through an intermediary set of equitable partitions; motivated by an analogy to stable marriages, we prove that the set of equitable partitions for a fixed word forms a distributive lattice when ordered component wise. We conclude that the general sweep maps defined in Armstrong et al. (Adv Math 284:159–185, 2015) are bijective. As a special case of particular interest, this gives the first proof that the zeta map on rational Dyck paths is a bijection.

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Authors and Affiliations

  1. LaCIM, Université du Québec à Montéal, Montréal, QC, Canada

    Hugh Thomas

  2. Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080-3021, USA

    Nathan Williams

Authors
  1. Hugh Thomas
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  2. Nathan Williams
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Correspondence to Nathan Williams.

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Thomas, H., Williams, N. Sweeping up zeta. Sel. Math. New Ser. 24, 2003–2034 (2018). https://doi.org/10.1007/s00029-018-0408-0

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  • DOI: https://doi.org/10.1007/s00029-018-0408-0

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