Explicit expressions for the moments of the size of an (s, s + 1)-core partition with distinct parts

In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the distribution is asymptotically normal. However, our analysis hinged on a characterization of (n,n+1)-core partitions given by Straub, which is not readily generalized to other families of simultaneous core partitions. In another paper (arXiv:1611.05775) with Doron Zeilberger, we made use of the characterization in terms of posets to analyze (2n+1,2n+3)-core partitions with distinct parts; here, the distribution was found not to be asymptotically normal. Our method involved finding recursive structure in the relevant sequence of posets. We remarked that this method is applicable to other families of core partitions, provided that one can understand the corresponding posets. Here, we use the poset method (and, as before, a computer) to analyze (n,dn-1)-core partitions with distinct parts, where d is a natural number. (This problem was introduced by Straub in arXiv:1601.07161.) We exhibit formulas for the moments of the size, as functions of d with n fixed, and vice versa. We conjecture that the distribution is asymptotically normal as n approaches infinity. Finally, we find formulas for the first few moments, as functions of both n and d.