Arithmetic properties and asymptotic formulae for \(\sigma_o \mathrm{mex}(n)\) and \(\sigma_e \mathrm{mex}(n)\) (Q6612388)

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\end{align*}\N\NIn the paper under review, the authors first prove the following two congruences modulo \(4\) for \(\sigma_e\operatorname{mex}(n)\) by utilizing some \(q\)-series identities:\N\begin{align*}\N\sigma_e\operatorname{mex}(10n+6) &\equiv0\pmod{4},\\\N\sigma_e\operatorname{mex}(10n+8) &\equiv0\pmod{4}.\N\end{align*}\NThey next establish some infinite families of congruences modulo \(4\) and \(8\) for \(\sigma_o\operatorname{mex}(n)\) and \(\sigma_e\operatorname{mex}(n)\). For example, for any \(n\geq0\),\N\[\N\sigma_e\operatorname{mex}{\left(2p^2n+kp+\dfrac{p^2-1}{12}\right)}\equiv0\pmod{4},\N\]\Nwhere \(p\) is a prime number satisfying \(p\equiv5,7,11\pmod{12}\), \(k\) is odd and \(1\leq k<p\).\N\NFinally, the authors deduce the following asymptotic formulas for \(\sigma_o\operatorname{mex}(n)\) and \(\sigma_e\operatorname{mex}(n)\) by applying Ingham s Tauberian theorem:\N\[\N\sigma_o\operatorname{mex}(n)\sim\sigma_e\operatorname{mex}(n)\sim\dfrac{1}{8\sqrt[4]{6n^3}} \exp{\left(\pi\sqrt{\dfrac{2n}{3}}\right)}.\N\]