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

Huan Xiong, Wenston J. T. Zang

Published 2018-04-24Version 1

Amdeberhan's conjectures on $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-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 moments of $X_{n,n+1}$ and $X_{2n+1,2n+3}$ were given by Zaleski and Zeilberger. Zaleski also studied the expectations and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we prove several polynomiality results for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper. 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. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves Zaleski's polynomiality conjecture on the $k$-th moment of $X_{n,dn-1}$. Moreover, when $d=1$, we show that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity.