Monotonicity Properties of N_(2)(m,n) and M_(2)(m, n)  

N_(2)(m,n)和M_(2)(m,n)的单调性

作　　者：QU Shuailong CUI Suping 曲帅龙;崔素平(青海师范大学数学与统计学院,青海西宁810008;高原科学与可持续发展研究院,青海西宁810008)

机构地区：[1]School of Mathematics and Statistics,Qinghai Normal University,Xining,Qinghai,810008,P.R.China [2]Academy of Plateau Science and Sustainability,Xining,Qinghai,810008,P.R.China

出　　处：《数学进展》2024年第5期983-992,共10页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.12001309);Natural Science Foundation Youth Fund of Qinghai Province(No.2022-ZJ-972Q)。

摘　　要：Let N_(2)(m,n)denote the number of partitions of n without repeated odd parts whose M_(2)-rank is m and M_(2)(m,n)denote the number of partitions π of n with distinct odd parts and such that the partition has crank m.In this paper,by means of some classical q-series identities,we prove some inequalities of N_(2)(m,n)and M_(2)(m,n).For example,for m≥4 and n≠2m+2,N_(2)(m,n)≤N_(2)(m,n+1).记N_(2)(m,n)表示M_2秩为m且没有重复奇数部分的n的分拆个数.同时,令M_(2)(m,n)表示n的分拆中π具有不同奇数部分,且π_(e)/2具有陪秩m的分拆的个数.本文利用一些经典的q-级数等式得到了许多关于N_(2)(m,n)与M_(2)(m,n)的不等式.例如,若m≥4并且n≠2m+2时,我们可以得到N_(2)(m,n)≤N_(2)(m,n+1).

关 键 词：PARTITION CRANK RANK 

分 类 号：O157.1[理学—数学]