The minimal excludant, or mex-function, on a set of positive integers is the smallest positive integer not in it. Andrews and Newman defined the mex-function mexA,a(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\text{ mex}_{A,a}(\lambda )$$\end{document} to be the smallest positive integer congruent to a modulo A that is not part of partition λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, and denote by pA,a(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{A,a}(n)$$\end{document} (reps. p¯A,a(n)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{p}_{A,a}(n)$$\end{document}) the number of partitions λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} of n satisfying mexA,a(λ)≡a(reps.a+A)(mod2A),\documentclass[12pt]{minimal}
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\begin{document}$$\text{ mex}_{A,a}(\lambda )\equiv a \text{(reps. } a+A{\text{) } }\pmod {2A},$$\end{document} and found numerous surprising identities involving these functions. Motivated by the above results, in this paper, we prove that the number of the partitions of n with an even (resp. odd) number of even parts equals the mex-function p4,2(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{4,2}(n)$$\end{document} (reps. p¯4,2(n)\documentclass[12pt]{minimal}
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\begin{document}$$\overline{p}_{4,2}(n)$$\end{document}). We also derive several identities connecting the differences of two mex-functions with partitions restricted by certain congruences, which develops the work of Dhar, Mukhopadhyay, and Sarma. Furthermore, we extend the mex-function to overpartitions and study the relevant properties.
