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 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] 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 E[X k n,n+1 ] 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 on the polynomiality of the expectation and higher moments of X n,dn−1 .

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

“…Also, simultaneous core partitions are connected with Motzkin paths and Dyck paths (see [9,11,39,40]). Some statistics of simultaneous core partitions, such as numbers of partitions, numbers of corners, largest sizes and average sizes, have attracted much attention in the past twenty years (see [2,3,4,7,8,10,13,14,15,18,20,21,23,24,25,29,31,33,35,37,42,43,44]). For example, Anderson [3] showed that the number of (s 1 , s 2 )-core partitions is equal to 1 s 1 +s 2 s 1 +s 2 s 1…”