Counting corners in partitions

被引:0
作者
Aubrey Blecher
Charlotte Brennan
Arnold Knopfmacher
Toufik Mansour
机构
[1] University of the Witwatersrand,The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics
[2] University of Haifa,Department of Mathematics
来源
The Ramanujan Journal | 2016年 / 39卷
关键词
Partitions; Generating functions; Corners; Asymptotics; Primary 05A15; 05A16;
D O I
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中图分类号
学科分类号
摘要
A partition of a positive integer n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} is a non-increasing sequence of positive integers whose sum is n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}. It may be represented by a Ferrers diagram. These diagrams contain corners which are points of degree two. We define corners of types (a,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)$$\end{document}, (a+,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a+,b)$$\end{document} and (a+,b+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a+,b+)$$\end{document}, and also define the size of a corner. Via a generating function, we count corners of each type and corners of size m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document}. We also find asymptotics for the number of corners as n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} tends to infinity.
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页码:201 / 224
页数:23
相关论文
共 6 条
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