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

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 (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 Xs,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 +1 ] and E[X 2+1,2+3 ] were given by Zaleski and Zeilberger (2017) when k is small. Zaleski (2017) also studied the expectation and higher moments of Xn,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 Xn,dn+1 and Xn,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 2k, when d is given and n tends to infinity. Moreover, when d = 1, we derive that the k-th moment E[X +1 ] of Xn,n+1 is asymptotically equal to (n2/10)k when n tends to infinity. The explicit formulas for the expectations E[Xn,dn+1] and E[Xn,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 Xn,dn−1.