On the distribution of rank and crank statistics for integer partitions

被引：0
作者：
Nian Hong Zhou
机构：
[1] East China Normal University,School of Mathematical Sciences
来源：
Research in Number Theory | 2019年 / 5卷
关键词：
Asymptotics; Partitions; Crank; Rank; Primary: 11P82; Secondary: 05A16; 05A17;
D O I：
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摘要：
Let k be a positive integer and m be an integer. Garvan’s k-rank Nk(n,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_k(n,m)$$\end{document} is the number of partitions of n into at least (k-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k-1)$$\end{document} successive Durfee squares with k-rank equal to m. In this paper we give some asymptotics for Nk(n,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_k(n,m)$$\end{document} with |m|≥n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|m|\ge \sqrt{n}$$\end{document} as n→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty .$$\end{document} As a corollary, we give a more complete answer for the Dyson’s crank distribution conjecture. We also establish some asymptotic formulas for finite differences of Nk(n,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_k(n,m)$$\end{document} with respect to m with m≫nlogn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\gg \sqrt{n}\log n.$$\end{document}
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