On the asymptotic distribution of cranks and ranks of cubic partitions

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• 作者：Byungchan Kim^a ; ^{bkim4@seoultech.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Eunmi Kim^b ; ^{ekim@kias.re.kr" class="auth_mail" title="E-mail the corresponding author} ; Hayan Nam^c ; ^{hayann@uci.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Asymptotics ; Cubic partitions ; Crank ; Rank
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：443
• 期：2
• 页码：1095-1109
• 全文大小：355 K

文摘

Cubic partitions are a special kind of bi-partitions whose name is inspired by a connection between a cubic continued fraction and an arithmetic property of this bi-partition. There are two partition statistics, namely rank and crank, for cubic partitions, which explain cubic partition congruences combinatorially. We obtain asymptotics for the number of cubic partitions of rank (resp. crank) m which reveal the distribution of the rank (resp. crank) among cubic partitions. As applications, we derive asymptotic inequalities between cubic partition rank and crank functions, and we remark upon the similarities and the differences between these and the corresponding inequalities for ordinary partitions.