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(201...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 ar Xiv: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 theβ-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).展开更多
基金supported by Swiss National Science Foundation(Grant No.P2ZHP2171879)。
文摘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 ar Xiv: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 theβ-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).
