Bisected theta series, least r-gaps in partitions, and polygonal numbers

Abstract

The least r-gap, \(g_r(\lambda )\), of a partition \(\lambda \) is the smallest positive integer that does not appear at least r times as a part of \(\lambda \). In this article, we introduce two new partition functions involving least r-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler’s partition function p(n), polygonal numbers, and the new partition functions. To prove the results, we use an interplay of combinatorial and q-series methods. We also give a combinatorial interpretation for

$$\begin{aligned} \sum _{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2). \end{aligned}$$

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Correspondence to Cristina Ballantine.

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This work was partially supported by a Grant from the Simons Foundation (#245997 to Cristina Ballantine).

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Ballantine, C., Merca, M. Bisected theta series, least r-gaps in partitions, and polygonal numbers. Ramanujan J 52, 433–444 (2020). https://doi.org/10.1007/s11139-018-0123-7

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Keywords

  • Partitions
  • Least gap
  • Polygonal numbers
  • Theta series

Mathematics Subject Classification

  • 05A17
  • 11P83