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Hook length inequalities for t-regular partitions in the t-aspect

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  24 July 2025

Gurinder Singh Open the ORCID record for Gurinder Singh [Opens in a new window]  and
Rupam Barman Open the ORCID record for Rupam Barman [Opens in a new window]
Gurinder Singh
Affiliation:
Department of Mathematics, https://ror.org/0022nd079 Indian Institute of Technology Guwahati , Guwahati, Assam 781039, India e-mail: gurinder.singh@iitg.ac.in
Rupam Barman*
Affiliation:
Department of Mathematics, https://ror.org/0022nd079 Indian Institute of Technology Guwahati , Guwahati, Assam 781039, India e-mail: gurinder.singh@iitg.ac.in
*
e-mail: rupam@iitg.ac.in
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Abstract

Let $t\geq 2$ and $k\geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t-regular partitions of n. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of k. We prove that for any $t\geq 2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq 0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. Finally, we state some problems for future works.

Keywords

Hook lengthst-regular partitionspartition inequalities

MSC classification

Primary: 11P81: Elementary theory of partitions 05A17: Partitions of integers 05A19: Combinatorial identities, bijective combinatorics 05A15: Exact enumeration problems, generating functions

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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