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BIASES IN INTEGER PARTITIONS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  14 January 2021

BYUNGCHAN KIM Open the ORCID record for BYUNGCHAN KIM [Opens in a new window]  and
EUNMI KIM Open the ORCID record for EUNMI KIM [Opens in a new window]
BYUNGCHAN KIM
Affiliation:
School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul1811, Republic of Korea e-mail: bkim4@seoultech.ac.kr
EUNMI KIM*
Affiliation:
Institute of Mathematical Sciences, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul03760, Republic of Korea
*
e-mail: ekim67@ewha.ac.kr
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Abstract

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$ . We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$ . We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$ .

Keywords

asymptotic formulabiaspartition

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 11P81: Elementary theory of partitions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 177 - 186
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

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-2019R1F1A1043415); Eunmi Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF–2020R1I1A1A01065877, NRF–2019R1A6A1A11051177).

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