THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

GOMEZ KEVIN; ROLEN LARRY; Joshua Males

doi:10.1017/S0004972722000764

THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

GOMEZ KEVIN
, ROLEN LARRY
, Joshua Males

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

5 Citations (Scopus)

Abstract

A number of recent papers have estimated ratios of the partition function p(n−j)/p(n) , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, f(j,n):=p(n)−2p(n−j)+p(n−2j) , and give another easy-to-use estimate of f(j,n) . As applications of these, we prove a shifted convexity property of p(n) , as well as giving new estimates of the k-rank partition function Nk(m,n) and non-k-ary partitions along with their differences.

Original language	English
Pages (from-to)	66-78
Number of pages	13
Journal	Bulletin of the Australian Mathematical Society
Volume	107
Issue number	1
Early online date	25 Aug 2022
DOIs	https://doi.org/10.1017/S0004972722000764
Publication status	Published - 1 Feb 2023

Bibliographical note

Funding Information:
The research of the second author conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021–2023 Dean’s Faculty Fellowship from Vanderbilt University and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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https://arxiv.org/abs/2203.11608v1

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title = "THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS",

abstract = "A number of recent papers have estimated ratios of the partition function p(n−j)/p(n) , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, f(j,n):=p(n)−2p(n−j)+p(n−2j) , and give another easy-to-use estimate of f(j,n) . As applications of these, we prove a shifted convexity property of p(n) , as well as giving new estimates of the k-rank partition function Nk(m,n) and non-k-ary partitions along with their differences.",

author = "GOMEZ KEVIN and ROLEN LARRY and Joshua Males",

note = "Funding Information: The research of the second author conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021–2023 Dean{\textquoteright}s Faculty Fellowship from Vanderbilt University and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. Publisher Copyright: {\textcopyright} The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.",

year = "2023",

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doi = "10.1017/S0004972722000764",

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AU - Males, Joshua

N1 - Funding Information: The research of the second author conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021–2023 Dean’s Faculty Fellowship from Vanderbilt University and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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N2 - A number of recent papers have estimated ratios of the partition function p(n−j)/p(n) , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, f(j,n):=p(n)−2p(n−j)+p(n−2j) , and give another easy-to-use estimate of f(j,n) . As applications of these, we prove a shifted convexity property of p(n) , as well as giving new estimates of the k-rank partition function Nk(m,n) and non-k-ary partitions along with their differences.

AB - A number of recent papers have estimated ratios of the partition function p(n−j)/p(n) , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, f(j,n):=p(n)−2p(n−j)+p(n−2j) , and give another easy-to-use estimate of f(j,n) . As applications of these, we prove a shifted convexity property of p(n) , as well as giving new estimates of the k-rank partition function Nk(m,n) and non-k-ary partitions along with their differences.

UR - https://doi.org/10.1017/S0004972722000764

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DO - 10.1017/S0004972722000764

M3 - Article (Academic Journal)

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JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 1

ER -

THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

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