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ON A CONJECTURE FOR A REFINEMENT OF THE SUM OF MINIMAL EXCLUDANTS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  20 July 2023

JULIA Q. D. DU  and
DAZHAO TANG Open the ORCID record for DAZHAO TANG [Opens in a new window]
JULIA Q. D. DU
Affiliation:
School of Mathematical Sciences, Hebei Workstation for Foreign Academicians, Hebei Normal University, Shijiazhuang 050024, PR China e-mail: qddu@hebtu.edu.cn
DAZHAO TANG*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, PR China
*
e-mail: dazhaotang@sina.com
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Abstract

In 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$, which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$. We confirm these two conjectures by using q-series manipulations and modular forms.

Keywords

congruencespartitionsminimal excludantmodular formsrefinement

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 276 - 287
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Julia Q. D. Du was partially supported by the National Natural Science Foundation of China (No. 12201177), the Hebei Natural Science Foundation (No. A2021205018), the Science and Technology Project of Hebei Education Department (No. BJK2023092), the Doctor Foundation of Hebei Normal University (No. L2021B02), the Program for Foreign Experts of Hebei Province and the Program for 100 Foreign Experts Plan of Hebei Province. Dazhao Tang was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ-MSX0387), the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509) and the Doctoral start-up research Foundation (No. 21XLB038) of Chongqing Normal University.

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