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On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts

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Abstract

Amdeberhan’s conjectures on the enumeration, the average size, and the largest size of (n,n+1)-core partitions with distinct parts have motivated many research on this topic. Recently, Straub (2016) and Nath and Sellers (2017) obtained formulas for the numbers of (n, dn − 1) and (n, dn+1)-core partitions with distinct parts, respectively. Let Xs,t be the size of a uniform random (s, t)-core partition with distinct parts when s and t are coprime to each other. Some explicit formulas for the k-th moments E[X kn,n+1 ] and E[X k2n+1,2n+3 ] were given by Zaleski and Zeilberger (2017) when k is small. Zaleski (2017) also studied the expectation and higher moments of Xn,dn−1 and conjectured some polynomiality properties concerning them in arXiv:1702.05634.

Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the k-th moments of Xn,dn+1 and Xn,dn−1 in this paper, by studying the β-sets of core partitions. In particular, we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2k, when d is given and n tends to infinity. Moreover, when d = 1, we derive that the k-th moment E[X kn,n+1 ] of Xn,n+1 is asymptotically equal to (n2/10)k when n tends to infinity. The explicit formulas for the expectations E[Xn,dn+1] and E[Xn,dn−1] are also given. The (n,dn−1)-core case in our results proves several conjectures of Zaleski (2017) on the polynomiality of the expectation and higher moments of Xn,dn−1.

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Acknowledgements

This work was supported by Swiss National Science Foundation (Grant No. P2ZHP2_171879). This work was done during the first author’s visit to the Harbin Institute of Technology (HIT). The first author thanks Professor Quanhua Xu and the second author for the hospitality. The authors really appreciate the helpful suggestions given by referees for improving the overall quality of the manuscript.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, Strasbourg, 67000, France

    Huan Xiong

  2. Institute of Advanced Study of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Wenston J. T. Zang

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  1. Huan Xiong
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  2. Wenston J. T. Zang
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Correspondence to Huan Xiong.

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Xiong, H., Zang, W.J.T. On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts. Sci. China Math. 64, 869–886 (2021). https://doi.org/10.1007/s11425-018-9500-x

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  • DOI: https://doi.org/10.1007/s11425-018-9500-x

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