Distinct r-tuples in integer partitions

被引：0

作者：

Archibald, Margaret ^{[1
,2
]}

Blecher, Aubrey ^{[1
,2
]}

Knopfmacher, Arnold ^{[1
,2
]}

机构：

[1] Univ Witwatersrand, John Knopfmacher Ctr Applicable Anal, Private Bag 3, ZA-2050 Johannesburg, South Africa

[2] Univ Witwatersrand, Number Theory Sch Math, Private Bag 3, ZA-2050 Johannesburg, South Africa

来源：

RAMANUJAN JOURNAL | 2019年 / 50卷 / 02期

基金：

新加坡国家研究基金会;

关键词：

Generating function; Integer partitions; r-tuples; NUMBER;

D O I：

10.1007/s11139-019-00180-x

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We define P-r(q) to be the generating functionwhich counts the total number of distinct (sequential) r -tuples in partitions of n and Q(r)(q, u) to be the corresponding bivariate generating function where u tracks the number of distinct r-tuples. These statistics generalise the number of distinct parts in a partition. In the early part of this paper we develop the tools by finding these generating functions for small cases r = 2 and r = 3. Then we use these methods to obtain P-r(q) and Q(r)(q, u) in the case of general r -tuples. These formulae are used to find the average number of distinct r-tuples for fixed r, as n -> infinity. Finally we show that as r -> infinity, q(-r) P-r(q) converges to an explicitly determined power series.

引用

页码：237 / 252

页数：16