Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman

Abstract

Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $\ddot{I}€$ of a positive integer $n$, $mex(\ddot{I}€)$, as the smallest positive integer that is not a part of $\ddot{I}€$. They defined $\ddot{I}ƒmex(n)$ to be the sum of $mex(\ddot{I}€)$ taken over all partitions $\ddot{I}€$ of $n$. We prove infinite families of congruence and multiplicative formulas for $\ddot{I}ƒmex(n)$. By restricting to the part of $\ddot{I}€$, Andrews and Newman also introduced $moex(\ddot{I}€)$ to be the smallest odd integer that is not a part of $\ddot{I}€$ and $\ddot{I}ƒmoex(n)$ to be the sum of $moex(\ddot{I}€)$ taken over all partitions $\ddot{I}€$ of $n$. In this article, we show that for any sufficiently large $X$, the number of all positive integer $n\le X$ such that $\ddot{I}ƒmoex(n)$ is an even (or odd) number is at least $\mathcal{O}(\log \log X)$.

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