Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Peak positions of strongly unimodal sequences


Authors: Kathrin Bringmann, Chris Jennings-Shaffer, Karl Mahlburg and Robert Rhoades
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 05A16; Secondary 11B83, 11F03, 11F12, 60C05
DOI: https://doi.org/10.1090/tran/7791
Published electronically: June 3, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05A16, 11B83, 11F03, 11F12, 60C05

Retrieve articles in all journals with MSC (2010): 05A16, 11B83, 11F03, 11F12, 60C05


Additional Information

Kathrin Bringmann
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: kbringma@math.uni-koeln.de

Chris Jennings-Shaffer
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: cjenning@math.uni-koeln.de

Karl Mahlburg
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: mahlburg@math.lsu.edu

Robert Rhoades
Affiliation: Susquehanna International Group, Bala Cynwyd, Pennsylvania 19004
Email: rob.rhoades@gmail.com

DOI: https://doi.org/10.1090/tran/7791
Keywords: Strongly unimodal sequences, unimodal rank, partition rank, Wright's circle method
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
Article copyright: © Copyright 2019 American Mathematical Society