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Partitions with fixed largest hook length

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Abstract

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.

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Acknowledgments

The authors thank Jiang Zeng for bringing Straub’s paper [18] to their attention and for some initial discussions that motivated this work. The authors also thank Zichen Yang for suggesting a better form of Theorem 2.7 and acknowledge the helpful suggestions made by the referee.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Huxi Campus LD301, Chongqing, 401331, People’s Republic of China

    Shishuo Fu & Dazhao Tang

Authors
  1. Shishuo Fu
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  2. Dazhao Tang
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Correspondence to Dazhao Tang.

Additional information

Both authors were supported by the Fundamental Research Funds for the Central Universities (No. CQDXWL-2014-Z004) and the National Science Foundation of China (No. 11501061).

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Fu, S., Tang, D. Partitions with fixed largest hook length. Ramanujan J 45, 375–390 (2018). https://doi.org/10.1007/s11139-016-9868-z

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  • DOI: https://doi.org/10.1007/s11139-016-9868-z

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