  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 $X_{n,dn-1}$.
