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Partitions with fixed differences between largest and smallest parts


Authors: George E. Andrews, Matthias Beck and Neville Robbins
Journal: Proc. Amer. Math. Soc. 143 (2015), 4283-4289
MSC (2010): Primary 11P84; Secondary 05A17
DOI: https://doi.org/10.1090/S0002-9939-2015-12591-9
Published electronically: April 2, 2015
MathSciNet review: 3373927
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Abstract: We study the number $ p(n,t)$ of partitions of $ n$ with difference $ t$ between largest and smallest parts. Our main result is an explicit formula for the generating function $ P_t(q) := \sum _{ n \ge 1 } p(n,t) \, q^n$. Somewhat surprisingly, $ P_t(q)$ is a rational function for $ t>1$; equivalently, $ p(n,t)$ is a quasipolynomial in $ n$ for fixed $ t>1$. Our result generalizes to partitions with an arbitrary number of specified distances.


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

  • [1] The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2014.
  • [2] George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067
  • [3] Philippe Flajolet and Robert Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge, 2009. MR 2483235
  • [4] Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112

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Additional Information

George E. Andrews
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: andrews@math.psu.edu

Matthias Beck
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
Email: mattbeck@sfsu.edu

Neville Robbins
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
Email: nrobbins@sfsu.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12591-9
Keywords: Integer partition, fixed difference between largest and smallest parts, rational generating function, quasipolynomial.
Received by editor(s): June 25, 2014
Published electronically: April 2, 2015
Additional Notes: The second author’s research was partially supported by the US National Science Foundation (DMS-1162638).
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society