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We show that the generating function of $\\sigma\\!\\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews \\textit{et al.} and Cohen. Further, we show that, as $n\\to \\infty$, $\\sigma\\!\\operatorname{maex}(n)$ is asymptotic to the sum of largest parts of all partitions of $n$. 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Let $\\sigma\\!\\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\\sigma\\!\\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews \\textit{et al.} and Cohen. Further, we show that, as $n\\to \\infty$, $\\sigma\\!\\operatorname{maex}(n)$ is asymptotic to the sum of largest parts of all partitions of $n$. 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