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

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 (2016) and Nath and Sellers (2017) 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 E[X n,n +1 k ] and E[X 2 n +1,2 n +3 k ] were given by Zaleski and Zeilberger (2017) when k is small. Zaleski (2017) 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 β -sets of core partitions. In particular, we show that these k -th moments are asymptotically some polynomials of n with degrees at most 2 k , when d is given and n tends to infinity. Moreover, when d = 1, we derive that the k -th moment E[X n,n +1 k ] of X n,n +1 is asymptotically equal to ( n 2 /10) k when n tends to infinity. The explicit formulas for the expectations E[X n,dn +1 ] and E[X n,dn −1 ] are also given. The ( n,dn −1)-core case in our results proves several conjectures of Zaleski (2017) on the polynomiality of the expectation and higher moments of X n,dn −1 .