Arithmetic properties and asymptotic formulae for \(\sigma_o \mathrm{mex}(n)\) and \(\sigma_e \mathrm{mex}(n)\)
type: Item
instance of: scholarly article
title: Arithmetic properties and asymptotic formulae for \(\sigma_o \mathrm{mex}(n)\) and \(\sigma_e \mathrm{mex}(n)\)
author: Rupam Barman
author: Gurinder Pal Singh
published in: The Ramanujan Journal
publication date: 2024-09-30T00:00:00Z
review text: A partition \(\lambda\) of a positive integer \(n\) is a weakly decreasing sequence of positive integers \(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r\) such that \(\sum_{i=1}^r\lambda_i=n\). The numbers \(\lambda_i\) are called the parts of the partition \(\lambda\). \textit{G. E. Andrews} and \textit{D. Newman} [Ann. Comb. 23, No. 2, 249--254 (2019; Zbl 1458.11011)] introduced the notion of minimal excludant of partitions. For a given partition \(\lambda\), the minimal excludant of \(\lambda\) is the least positive integer missing from \(\lambda\). Let \(\operatorname{mex}(\lambda)\) denote the minimal excludant of the partition \(\lambda\). Andrews and Newmann further considered another arithmetic function \(\sigma\operatorname{mex}(n)\), which is defined by\N\[\N\sigma\operatorname{mex}(n)=\sum_{\lambda\in\mathcal{P}(n)}\operatorname{mex}(\lambda),\N\]\Nwhere \(\mathcal{P}(n)\) denotes the set of all partitions of \(n\). Later, \textit{N. D. Baruah}, \textit{S. C. Bhoria}, \textit{P. Eyyunni} and \textit{B. Maji} [Ramanujan J. 62, No. 4, 1045--1067 (2023; Zbl 07784980)] refined the function \(\sigma\operatorname{mex}(n)\) by considering the sum of odd and even minimal excludants separately. More specifically, for any \(n\geq1\), they defined the following two arithmetic functions:\N\begin{align*}\N\sigma_o\operatorname{mex}(n) &:=\sum_{\substack{\lambda\in\mathcal{P}(n)\\\N\operatorname{mex}(\lambda) \equiv1\bmod{2}}}\operatorname{mex}(\lambda),\\\N\sigma_e\operatorname{mex}(n) &:=\sum_{\substack{\lambda\in\mathcal{P}(n)\\\N\operatorname{mex}(\lambda) \equiv0\bmod{2}}}\operatorname{mex}(\lambda).\N\end{align*}\NBaruah et al. also established three congruences modulo \(4\) and \(8\) for \(\sigma_o\operatorname{mex}(n)\) and \(\sigma_e\operatorname{mex}(n)\), namely, for any \(n\geq0\),\N\begin{align*}\N\sigma_o\operatorname{mex}(2n+1) &\equiv0\pmod{4},\\\N\sigma_o\operatorname{mex}(4n+1) &\equiv0\pmod{8},\\\N\sigma_e\operatorname{mex}(4n) &\equiv0\pmod{4}.\N\en…
reviewed by: Dazhao Tang
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 7920322
zbMATH Keywords: partitions
zbMATH Keywords: minimal exludents
zbMATH Keywords: congruences
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
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DOI: 10.1007/S11139-024-00886-7
description: scientific article; zbMATH DE number 7920322