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

Huan Xiong; Wenston J. T. Zang

paper

On the polynomiality and asymptotics of moments of sizes for random $(n, d n \pm 1)$ -core partitions with distinct parts

arXiv:1804.09091

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 and Nath-Sellers obtained formulas for the numbers of $(n, d n - 1)$ and $(n, d n + 1)$ -core partitions with distinct parts, respectively. Let $X_{s, t}$ 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 $E [X_{n, n + 1}^{k}]$ and $E [X_{2 n + 1, 2 n + 3}^{k}]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n, d n - 1}$ 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 $X_{n, d n + 1}$ and $X_{n, d n - 1}$ in this paper, by studying the beta sets of core partitions. In particular, we show that these $k$ -th moments are asymptotically some polynomials of n with degrees at most $2 k$ , when $d$ is given and $n$ tends to infinity. Moreover, when $d = 1$ , we derive that the $k$ -th moment $E [X_{n, n + 1}^{k}]$ of $X_{n, n + 1}$ is asymptotically equal to $(n^{2} /10)^{k}$ when $n$ tends to infinity. The explicit formulas for the expectations $E [X_{n, d n + 1}]$ and $E [X_{n, d n - 1}]$ are also given. The $(n, d n - 1)$ -core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n, d n - 1}$ .

This paper has been accepted for publication in SCIENCE CHINA Mathematics

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

Abstract

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On the polynomiality and asymptotics of moments of sizes for random $(n, d n \pm 1)$ -core partitions with distinct parts