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

TL;DR

This paper investigates the moments of sizes of random core partitions with distinct parts, establishing polynomiality and asymptotic formulas, and confirms several conjectures related to these properties.

Contribution

It derives polynomiality results and asymptotic formulas for moments of core partitions with parameters $(n, dn\uplus 1)$, confirming conjectures and extending prior formulas.

Findings

Moments are asymptotically polynomial in n with degree at most 2k.

For $d=1$, the k-th moment of $X_{n,n+1}$ behaves like $(n^2/10)^k$ as n grows.

Explicit formulas for expectations of sizes of core partitions are provided.

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 $(n, dn-1)$ 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 the $k$-th moments $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation 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 derive several polynomiality results and asymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ 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 $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive 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. 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 several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$.