Partitions weighted by the number of two types of parts

Byungchan Kim; Eunmi Kim

doi:10.4134/BKMS.b230752

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2024; 61(6): 1677-1684

Online first article August 19, 2024 Printed November 30, 2024

https://doi.org/10.4134/BKMS.b230752

Partitions weighted by the number of two types of parts

Byungchan Kim; Eunmi Kim

Seoul National University of Science and Technology; Ewha Womans University

Abstract

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Abstract

We confirm the conjecture proposed by ourselves and J. Lovejoy that for all $n>9$ \[ p'_e(n) > p'_o(n) \] holds, where $p'_e(n)$ (respectively, $p'_o(n)$) is the number of partitions of $n$ having an even (respectively, odd) number of odd parts larger than twice of the number of even parts. Moreover, we examine the connections between the number of partitions weighted by the number of two types of parts and partition functions from the literature on the theory of partitions.

Keywords: Integer partition, overpartition, positivity, partition injection

MSC numbers: Primary 05A17, 11P81

Supported by: Byungchan Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF--2022R1F1A1063530). Eunmi Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2023-00244423, NRF--2019R1A6A1A11051177).