On a Study of Some Analogues of Minimal Excludant

Abstract

This thesis primarily studies an interesting partition statistic named as `minimal excludant' or ``mex'' function. The minimal excludant, mex$(\pi)$ of an integer partition $\pi$, is the smallest positive integer that is not a part of $\pi$. The average size of this smallest gap of a partition was studied by Grabner and Knopfmacher \cite{grabner}. In $2019$, Andrews and Newman \cite{Andrews2019} explored the idea of minimal excludant where they define \begin{equation*}\label{sigmamex} \sigma\textup{mex}(n) = \sum_{\pi \in \mathcal{P}(n)} \textup{mex}(\pi), \end{equation*} where $\mathcal{P}(n)$ denote the collection of all integer partitions of $n$. We study the questions raised by Andrews and Newman for the function $\sigma \textup{mex}(n)$, but restricted to partitions into distinct parts. We define the function $\sigma_d\textup{mex}(n)$ by \begin{equation*}\label{sigma_d defn} \sigma_d \textup{mex}(n) = \sum_{\pi \in \mathcal{D}(n)} \textup{mex}(\pi), \end{equation*} where $\mathcal{D}(n)$ denote the collection of partitions of $n$ into distinct parts and interestingly we observe that it has a nice connection with Ramanujan's function $\sigma(q)$. We also define \begin{equation*} a_d(n)= \sum_{ \pi \in \mathcal{D}(n) \atop \textup{mex}(\pi) \, \textup{odd}} 1. \end{equation*} In fact, the generating function for $a_d(n)$ was considered by Uncu \cite{Uncu2019} in a different combinatorial context. Hence, as an application, we derive a stronger version of a result of Uncu.

In addition to the above mentioned work, a natural continuation to the study of minimal excludants is the study of second minimal excludant which we define as the second smallest integer missing from an integer partition $\pi$ and it is denoted by $\textup{mex}_2(\pi)$. We derive the generating function for $\sigma_2 \textup{mex}(n)$, where $\sigma_2 \textup{mex}(n) = \sum_{\pi \in \mathcal{P}(n)} \textup{mex}_2(\pi)$ and along with it we also study partitions with a fixed difference between the minimal excludant and the second minimal excludant. For this, we define $\Delta_t(n)$, the number of partitions $\pi$ of $n$ with $\textup{mex}_2(\pi) - \textup{mex}(\pi) = t$. We derive its generating function and as special cases, we obtain interesting identities connecting $\Delta_t(n)$ to $\sigma \textup{mex}(n)$ and certain restricted partition functions. This also leads us to the notion of a mex sequence and further we derive two neat identities involving the number of partitions whose mex sequence has length at least $r$.

Lastly, we study some more results on mex function which includes the study of generating function for $\sigma \textup{meex}(n)$, defined as the sum of the smallest even integers that are missing in all the partitions of $n$. We also study the sum of minimal even excludant in distinct part partitions and thus found a combinatorial identity. We further study the sum of squares of minimal excludant, denoted by $\sigma \textup{mex}^2 (n)$ and established its relation with mex functions.