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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Peak positions of strongly unimodal sequences
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by Kathrin Bringmann, Chris Jennings-Shaffer, Karl Mahlburg and Robert Rhoades PDF
Trans. Amer. Math. Soc. 372 (2019), 7087-7109 Request permission

Abstract:

We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and we prove an asymptotic result in support of this conjecture. Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and we prove that its values (when appropriately renormalized) are normally distributed with mean $0$ in the asymptotic limit.
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Additional Information
  • Kathrin Bringmann
  • Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
  • MR Author ID: 774752
  • Email: kbringma@math.uni-koeln.de
  • Chris Jennings-Shaffer
  • Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
  • MR Author ID: 1061334
  • Email: cjenning@math.uni-koeln.de
  • Karl Mahlburg
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
  • MR Author ID: 664593
  • Email: mahlburg@math.lsu.edu
  • Robert Rhoades
  • Affiliation: Susquehanna International Group, Bala Cynwyd, Pennsylvania 19004
  • MR Author ID: 762187
  • Email: rob.rhoades@gmail.com
  • Received by editor(s): November 22, 2018
  • Published electronically: June 3, 2019
  • Additional Notes: The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant agreement no. 335220—AQSER
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7087-7109
  • MSC (2010): Primary 05A16; Secondary 11B83, 11F03, 11F12, 60C05
  • DOI: https://doi.org/10.1090/tran/7791
  • MathSciNet review: 4024548