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Amdeberhan's conjectures on the enumeration, the average size, and thelargest size of $(n,n+1)$-core partitions with distinct parts have motivatedmany research on this topic. Recently, Straub and Nath-Sellers obtainedformulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions withdistinct 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 eachother. 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 inarXiv:1702.05634.  Motivated by the above works, we derive several polynomiality results andasymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ inthis paper, by studying the beta sets of core partitions. In particular, weshow that these $k$-th moments are asymptotically some polynomials of n withdegrees 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 $n$ tendsto infinity. The explicit formulas for the expectations $\mathbb{E}[X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-corecase in our results proves several conjectures of Zaleski on the polynomialityof the expectation and higher moments of $X_{n,dn-1}$.

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