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

• 作者: Huan Xiong Wenston J.T.Zang
• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2021,（4）
• 所属期刊栏目: Articles
• 页码范围: 869-886
• 页数: 18页
• 语种: 英文

摘要

Amdeberhan's conjectures on the enumeration,the average size,and the largest size of (n,n+1)-core partitions with distinct parts have motivated much 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[Xnk,n+1]and E[Xk2n+1,2n+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[Xnk,n+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.