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\begin{center}
\vskip 1cm{\LARGE\bf 
The Minimal Excludant in Integer Partitions
}
\vskip 1cm
\large
George E. Andrews \\
The Pennsylvania State University\\
University Park, PA 16802\\
USA\\
\href{mailto:gea1@psu.edu}{\tt gea1@psu.edu}\\
\ \\
David Newman\\
Far Rockaway, New York\\
USA \\
\href{mailto:DavidSNewman@gmail.com}{\tt DavidSNewman@gmail.com} 
\end{center}

\vskip .2 in
\begin{abstract}
The minimal excludant, or ``$\mex$'' function, on a set $S$ of positive
integers is the least positive integer not in $S$. In this paper, the mex
function is extended to integer partitions generalized by constricting the
universal set from all positive integers to those in certain arithmetic
progressions. There are numerous surprising partition identities connected
with this restricted mex function. This paper provides an account of
some of the most conspicuous cases.
\end{abstract}

\section{Introduction}\label{Intro}

The minimal excludant function (mex-function) appears extensively
in combinatorial game theory (e.g., \cite{8}).
For each set $S$ of nonnegative integers we
define it as follows:
\begin{equation} \label{1.1}
\MEX(S)=\min(\integers_{\geq 0}\backslash S).
\end{equation}
The mex function is perhaps best known in its uses in the game of nim
and the Sprague-Grundy theory originating in the 1930's.

We shall generalize the mex-function to apply to integer partitions. We
define $\mex_{A,a}(\pi)$ to be the smallest integer congruent to $a$
modulo $A$ that is not a part of the integer partition $\pi$.

We define $p_{A,a}(n)$ to be the number of partitions $\pi$ of $n$, where
\begin{equation}
\label{1.2} \mex_{A,a}(\pi)\equiv a \pmod{2A},
\end{equation}
and $\overline{p}_{A,a}(n)$ to be the number of partitions $\pi$ of $n$, where 
\begin{equation}
\label{1.3} \mex_{A,a}(\pi)\equiv A+a\pmod{2A}.
\end{equation}
If $p(n)$ denotes the number of integer partitions of $n$, then clearly
\begin{equation}
\label{1.4} p(n)=p_{A,a}(n)+\overline{p}_{A,a}(n).
\end{equation}

For example, consider $n=4$, $A=2$, and $a=1$. There are five partitions of 4, so $p(4)=5$, where $p(n)$ is the number of partitions of $n$. These partitions along with first missing odd part (m.o.) are 4 (m.0.~1), $3+1$ (m.o.~5),
$2+2$ (m.o.~1), $2+1+1$ (m.o.~3) and $1+1+1+1$ (m.o.~3). Thus three partitions have $\mex_{2,1}$ congruent to 1 mod 4,
and two have $\mex_{2,1}$ congruent to 3 mod 4.
Thus $p_{2,1}(4)=3$ and $\overline{p}_{2,1}(4)=2$.

Let us define
\begin{equation}
\label{1.5} F_{A,a}(q)=\sum_{n\geq 0} p_{A,a}(n)q^n,
\end{equation}
and
\begin{equation}
\label{1.6} \overline{F}_{A,a}(q)=\sum_{n\geq 0} \overline{p}_{A,a}(n)q^n .
\end{equation}
Then
\begin{align}
\label{1.7} F_{A,a}(q)+\overline{F}_{A,a}(q)&=\sum_{n\geq 0} p(n)q^n\\
\notag & = \prod_{n=1}^\infty \frac{1}{1-q^n}:=F(q),\text{ by \cite[p.\ 3, Th.\ 1.1]{1}.}
\end{align}
Thus by \eqref{1.7}, every identity for $F_{A,a}(q)$ yields a natural identity for $\overline{F}_{A,a}(q)$ and vice versa.

Our theorems also require $p_e(n)$ (resp., $p_o(n)$), the number of partitions of $n$ into an even (resp., odd) numbers of parts. 
Hence by the methods of \cite[Ch.\ 1]{1}, we get
\begin{equation}
\label{1.8} F_e(q):=\sum_{n\geq 0} p_e(n)q^n=\sum_{n=0}^\infty \frac{q^{2n}}{(q;q)_{2n}},
\end{equation}
and
\begin{equation}
\label{1.9} F_o(q):=\sum_{n\geq 0} p_o(n)q^n=\sum_{n=0}^\infty \frac{q^{2n+1}}{(q;q)_{2n+1}},
\end{equation}
where
\begin{equation}
\label{1.10} (x;q)_n=(1-x)(1-xq)\cdots(1-xq^{n-1}).
\end{equation}

In order to prove our theorems for $p_{1,1}(n)$ and $p_{3,3}(n)$, we must recall two partition statistics, the rank and the crank.

The \emph{rank} of a partition is the largest part minus the number of parts. The \emph{crank} of a partition is the largest part of the partition if there are no ones as parts, and otherwise is the number of parts larger than the number of ones minus the numbers of ones. 

Our first theorem (which is actually a lemma for treating $p_{1,1}(n)$) is originally due to Uncu \cite{11}. Our proof differs from his and is included for completeness. Also see Somos \cite{10}.
\begin{theorem}\label{Theorem1} The generating function for partitions with non-negative crank is
\begin{equation}
\label{1.11} \frac{1}{(q;q)_\infty}\sum_{n\geq 0} (-1)^n q^{\binom{n+1}{2}}.
\end{equation}
\end{theorem}
This leads directly to the following three results:
\begin{theorem}\label{Theorem2} $p_{1,1}(n)$ equals the number of partitions of $n$ with non-negative crank.\end{theorem}

\begin{theorem}\label{Theorem3} $p_{3,3}(n)$ equals the number of partitions of $n$ with rank $\geq -1$.\end{theorem}

\begin{theorem}\label{Theorem4} $p_{2,1}(n)=p_e(n)$.\end{theorem}

Our last two theorems are the real surprises in this study. The theory of partitions is replete with uncanny partition identities, of which the first Rogers-Ramanujan identity is the most famous example:
\begin{theorem}[Rogers-Ramanujan] \cite[Ch.\ 7]{1} 
The number of partitions of $n$ in which the difference between parts
is at least 2 equals the number of partitions of $n$ in which all parts
are congruent to 1 or 4 modulo 5.
\end{theorem}

It turns out that $p_{4,2}(n)$ and $p_{6,3}(n)$ appear in identities of this nature. 

\begin{theorem}\label{Theorem5} $p_{4,2}(n)-p_o(n)$ equals the number of
partitions of $n$ into parts congruent to $\pm 4, \pm 6, \pm 8, \pm 10$
{\rm (mod} $32${\rm )}.\end{theorem}

\begin{theorem} \label{Theorem6} $p_{6,3}(n)-p_o(n)$ equals the number
of partitions of $n$ into parts congruent to $\pm 2, \pm 4, \pm 5, \pm 6,
\pm 7, \pm 8$ {\rm (mod} $24${\rm )}.\end{theorem}

Table~\ref{tab1} gives the first few values of these sequences.
\begin{table}[H]
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
$n$ & $p_{11}(n)$ & $p_{33}(n)$ & $p_{21}(n)$ & $p_{42}(n)$ & $p_{63}(n)$ \\
\hline
0 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 1 \\
2 & 1 & 2 & 1 & 1 & 2 \\
3 & 2 & 2 & 1 & 2 & 2 \\
4 & 3 & 4 & 3 & 3 & 4 \\
5 & 4 & 5 & 3 & 4 & 5 \\
6 & 6 & 8 & 6 & 6 & 8 \\
7 & 8 & 10 & 7 & 8 & 10 \\
8 & 12 & 15 & 12 & 12 &15 \\
9 & 16 & 20 & 14 & 16 & 19 \\
10 & 23 & 28 & 22 & 22 & 27 \\
11 & 30 & 36 & 27 & 29 & 34 \\
12 & 42 & 50 & 40 & 40 & 48 \\
13 & 54 & 64 & 49 & 52 & 60 \\
14 & 73 & 86 & 69 & 69 & 81 \\
15 & 94 & 110 & 86 & 90 & 102 \\
16 & 124 & 145 & 118 & 118 & 135 \\
17 & 158 & 184 & 146 & 151 & 169 \\
18 & 206 & 238 & 195 & 195 & 220 \\
19 & 260 & 300 & 242 & 248 & 224 \\
20 & 334 & 384 & 317 & 317 & 352 \\
21 & 419 & 481 & 392 & 400 & 437 \\
22 & 531 & 608 & 505 & 505 & 554 \\
23 & 662 & 756 & 623 & 632 & 684 \\
24 & 832 & 948 & 793 & 793 & 860 \\
25 & 1029 & 1172 & 973 & 985 & 1057 
\end{tabular}
\end{center}
\caption{First few values of $p_{1,1}$, $p_{33}(n)$, $p_{21}(n)$,
$p_{42}(n)$, and $p_{63}(n)$}
\label{tab1}
\end{table}
We note that $p_{1,1}(n)$ is sequence \seqnum{A064428} in the 
{\it On-Line Encyclopedia of Integer Sequeces} (OEIS),
$p_{3,3}(n)$ is \seqnum{A260894}, $p_{2,1}(n)$ is \seqnum{A027187},
and $p_{4,2}(n)$ is \seqnum{A046682}. 
We also note that none of the references to these sequences in the OEIS
refer to our results.

We note that the proof of Theorem~\ref{Theorem3} relies on the study of Garden of Eden partitions given by Hopkins and Sellers \cite{9},
while Theorem~\ref{Theorem5} and \ref{Theorem6} rely on identities given by Blecksmith, Brillhart and Gerst \cite{6}.

\section{Uncu's theorem} \label{Sec2}

If $M(m,n)$ denotes the number of partitions of $n$ with crank $m$, then \cite{5}
\begin{equation}
\label{2.1} \sum_{n=0}^\infty \sum_{m=-\infty}^\infty M(m,n) z^mq^n=\frac{(q;q)_\infty}{(zq;q)_\infty(z^{-1}q;q)_\infty}.
\end{equation}
Now by \cite[p.~19, Eq.~(2.2.5)]{1} we have
\begin{equation}
\label{2.2} \frac{(q;q)_\infty}{(zq;q)_\infty(z^{-1}q;q)_\infty}=(q;q)_\infty \sum_{m=0}^\infty \frac{z^mq^m}{(q;q)_m}\sum_{n=0}^\infty \frac{z^{-n}q^n}{(q;q)_n}.
\end{equation}

Thus the generating function for partitions with non-negative crank consists of those terms on the right-hand side of \eqref{2.2}, where the exponent on $z$ is non-negative. Hence the required generating function is
\begin{align*}
(q;q)_\infty\sum_{n=0}^\infty \sum_{m=n}^\infty \frac{q^{m+n}}{(q;q)_m(q;q)_n}
&=  (q;q)_\infty \sum_{n=0}^\infty\sum_{m=0}^\infty \frac{q^{m+2n}}{(q;q)_{m+n}(q;q)_n}\\
&=  (q;q)_\infty \sum_{m=0}^\infty \frac{q^m}{(q;q)_m} \sum_{n=0}^\infty \frac{q^{2n}}{(q;q)_n(q^{m+1};q)_n}\\
&=  (q;q)_\infty \sum_{m=0}^\infty \frac{q^m}{(q;q)_m(q^2;q)_\infty(q^{m+1};q)_\infty}\\
&\quad \times\sum_{n=0}^\infty \frac{(-1)^n q^{\binom{n+1}{2}+mn}(q^2;q)_n}{(q;q)_n}\\
& \quad \text{(by \cite[p.~19, Cor.~2.3, $a=b=0$, $c=q^{m+1}$, $t=q^2$]{1})}\\
&=  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{\binom{n+1}{2}}(1-q^{n+1})\sum_{m=0}^\infty q^{m(n+1)}\\
&=  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{\binom{n+1}{2}}.
\end{align*}
 
 
 \section{Background lemmas}\label{Sec3}
 
 \begin{lemma} \label{Lemma7}
 \begin{equation}
 \label{3.1} F_{2k,k}(q)=\frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{kn^2}.
 \end{equation}
 \end{lemma}
 
 \begin{proof}
 \begin{align*}
  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^nq^{kn^2}
& =  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty q^{4kn^2}(1-q^{4kn+k})\\
& =  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty q^{k+3k+5k+\cdots+k(4n-1)}(1-q^{k(4n+1)})\\
& =  \sum_{n=0}^\infty \frac{q^{k+3k+5k+\cdots+k(4n-1)}}{\displaystyle \prod_{\substack{m=1\\m\neq k(4n+1)}}^\infty (1-q^m)},
 \end{align*}
 and thus this last expression is clearly the generating function for $p_{2k,k}(n)$.
 \end{proof}

\begin{lemma}\label{Lemma8}
\begin{equation}
\label{3.2} F_{k,k}(q)  = \frac{1}{(q;q)_\infty}\sum_{n=0}^\infty (-1)^n q^{k\binom{n+1}{2}}.
\end{equation}
\end{lemma}

\begin{proof}
\begin{align*}
 \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{k\binom{n+1}{2}}
&=  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty q^{k\binom{2n+1}{2}}(1-q^{k(2n+1)})\\
&=  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty q^{k+2k+\cdots+2nk}(1-q^{k(2n+1)})\\
&=  \sum_{n=0}^\infty \frac{q^{k+2k+\cdots+2nk}}{\displaystyle \prod_{\substack{m=1\\m\neq k(2n+1)}}^\infty (1-q^m)},
\end{align*}
and this last expression is clearly the generating function for $p_{k,k}(n)$.
\end{proof}

\section{Proof of Theorem~\ref{Theorem2}}\label{Sec4}

\begin{proof}
\begin{align}
\label{4.1}  \sum_{n=0}^\infty p_{1,1}(n)q^n
\notag & =  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{\binom{n+1}{2}} \qquad\text{(by Lemma \ref{Lemma8} with $k=1$)}
\end{align}
and by Theorem~\ref{Theorem1}, this is the generating function for partitions with non-negative crank. 
\end{proof}

\section{Proof of Theorem~\ref{Theorem3}}\label{Sec5}

\begin{proof}
In \cite{9}, Hopkins and Sellers prove that Garden of Eden partitions are equinumerous with partitions whose rank is $\leq -2$. They then quote the work of Dyson \cite{7} to reveal that this generating function is
$$
\frac{1}{(q;q)_\infty} \sum_{n=1}^\infty (-1)^{n-1}q^{3\binom{n+1}{2}}.
$$
Hence
$$
F_{3,3}(q)=\frac{1}{(q;q)_\infty}\sum_{n=0}^\infty (-1)^n q^{3\binom{n+1}{2}}=\frac{1}{(q;q)_\infty} \left(1-\sum_{n=1}^\infty (-1)^{n-1}q^{3\binom{n+1}{2}}\right)
$$
and this generates all partitions excluding those with rank $\leq -2$, i.e.,\ partitions with rank $\geq -1$. 
\end{proof}

\section{Proof of Theorem~\ref{Theorem4}}\label{Sec6}
\begin{proof}
\begin{align*}
F_{2,1}(q) &=  \frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{n^2}\\
&=  \frac{1}{2(q;q)_\infty}\left(\sum_{n=-\infty}^\infty (-1)^n q^{n^2}+1\right)\\
&=  \frac{1}{2(q;q)_\infty}\left(\frac{(q;q)_\infty}{(-q;q)_\infty}+1\right)\\
&=  \frac{1}{2}\left(\frac{1}{(-q;q)_\infty}+\frac{1}{(q;q)_\infty}\right)\\
&=  \frac{1}{2}\left(\sum_{n=0}^\infty\frac{(-1)^nq^n}{(q;q)_n}+\sum_{n=0}^\infty\frac{q^n}{(q;q)_n}\right)\\
&=  \sum_{n=0}^\infty \frac{q^{2n}}{(q;q)_{2n}}\\
&=  \sum_{n\geq 0} p_e(n)q^n.
\end{align*}
\end{proof}

\section{Proof of Theorem~\ref{Theorem5}}\label{Sec7}

\begin{proof}
In the work of Blecksmith, Brillhart and Gerst \cite[Theorem 1 (a)]{6}, we find
\begin{equation}
\label{7.1} \prod_{\substack{n=1\\n\not\equiv \pm 4, \pm 6, \pm 8, \pm 10\text{ (mod }32)}}^\infty (1-q^n)=1+\sum_{n=1}^\infty (-1)^n\left(q^{n^2}+q^{2n^2}\right).
\end{equation}
Let us now divide both sides of \eqref{7.1} by $(q;q)_\infty$, and we find
\begin{align}
\label{7.2} \prod_{\substack{n=1\\n\equiv \pm 4, \pm 6, \pm 8, \pm 10 \text{ (mod }32)}}^\infty \frac{1}{1-q^n} &=  F_{2,1}(q)+F_{4,2}(q)-\frac{1}{(q;q)_\infty}\\
\notag &=  F_{4,2}(q)-\overline{F_{2,1}}(q) & \text{(by \eqref{1.7})}\\
\notag & = F_{4,2}(q)- \sum_{n=0}^\infty p_o(n)q^n, & \text{(by Theorem~\ref{Theorem4} and \eqref{1.7})}
\end{align}
and this is equivalent to Theorem~\ref{Theorem5}.
\end{proof}

\section{Proof of Theorem~\ref{Theorem6}}\label{Sec8}

\begin{proof}
Again, we turn to the paper of Blecksmith, Brillhart and Gerst, where we find \cite[Theorem 3 (a)]{6}
\begin{align}
\label{8.1}  \prod_{\substack{n=1\\n\not\equiv \pm 2, \pm 4, \pm 5, \pm 6, \pm 7, \pm 8 \text{ (mod }24)}}^\infty (1-q^n)
& =  1 + \sum_{n=1}^\infty (-1)^n\left(q^{n^2}+q^{3n^2}\right).
\end{align}
We now divide by $(q;q)_\infty$ to obtain
\begin{align}
\label{8.2}  \prod_{\substack{n=1\\n\equiv \pm 2, \pm 4, \pm 6, \pm 7, \pm 8\text{ (mod }24)}} \frac{1}{1-q^n}& =  F_{2,1}(q)+F_{6,3}(q)-\frac{1}{(q;q)_\infty}\\
\notag &=  F_{6,3}(q)-\overline{F_{2,1}}(q) & \text{(by \eqref{1.7})}\\
\notag &=  F_{6,3}(q)-\sum_{n=0}^\infty p_o(n) q^n, & \text{(by Theorem~\ref{Theorem4} and \eqref{1.7})}
\end{align}
and this is equivalent to Theorem~\ref{Theorem6}. 
\end{proof}

\section{Conclusion}

We would suggest that this is the tip of the iceberg in the study of $\mex_{A,a}(\pi)$. It is our hope that there are many interesting results awaiting further inquiry.

Among the more obvious questions are these:
\begin{enumerate}[I.]
\item Are there bijective proofs of any of these theorems?
\item Is there a theorem for $p_{2,2}(n)$ comparable to Theorem~\ref{Theorem2} and \ref{Theorem3}?

 To study $p_{2,2}(n)$, we would need to know more about 
 $$
 F_{2,2}(q)=\frac{1}{(q;q)_\infty} \sum_{n=0}^\infty (-1)^n q^{n^2+n}.
 $$
 Utilizing the weak form of Bailey's lemma \cite[eqs.\ (3.27) and (3.33) with $a=q$, $\alpha_r=(-1)^n$]{2} we see that 
 $$
 F_{2,2}(q)=\sum_{n\geq 0} \frac{q^{n^2+n}B_n(q)}{(q;q)_{2n+1}},
 $$
 where
 $$
 B_n(q) = \sum_{r=0}^n (-1)^r \genfrac{[}{]}{0pt}{}{2n+1}{n-r}
 $$
 with the $q$-binomial coefficient given by 
 $$
 \genfrac{[}{]}{0pt}{}{m}{n}=\frac{(q;q)_m}{(q;q)_n(q;q)_{m-n}}.
 $$
 It appears that $B_n(q)$
 \begin{enumerate}[(1)]
 \item has nonnegative coefficients;
 \item is unimodal; and
 \item enumerates some subset of the partitions into at most $n$ parts each $\leq n+1$.
 \end{enumerate}
 We are unable to prove any of these assertions.

\item Are there comparable theorems for other $F_{A,a}(q)$?

We would note that in terms of classical theta functions
$$
F_{2k,k}(q) = \frac{1}{2(q;q)_\infty}(\theta_4(0,q^k)-1).
$$
Thus one can insert $F_{2k,k}(q)$ into classical modular equations such as \cite[p.\ 289, penultimate equation]{11}
\begin{equation}
\label{9.1} \psi(q^2)\theta_4(0,q^5)+q\psi(q^{10})\theta_4(0,q)=(-q;q)_\infty(-q^5;-q^5)_\infty;
\end{equation}
where \cite[p.\ 23, Eq.\ (2.2.13)]{1}
$$
\psi(q)=\sum_{n=0}^\infty q^{n(n+1)/2}=\frac{(q^2;q^2)_\infty}{(q;q^2)_\infty}.
$$
However, such complex identities do not appear to yield appealing partition identities comparable to any of Theorems~\ref{Theorem1} to \ref{Theorem6}.

\item Are there other partition-theoretic objects that are related to instances of $p_{A,a}(n)$?

With regard to IV., we note that in \cite[Sec.\ 5]{3} it is proved that $p_{1,1}(n)$ is also the number of concatenated spiral self-avoiding walks with an odd number of turns.
\end{enumerate}

\begin{thebibliography}{10}

\bibitem{2}
George~E. Andrews, {\em {$q$}-Series: Their Development and Application in
  Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra},
  Vol.~66 of {\em CBMS Regional Conference Series in Mathematics}, 
  American Mathematical Society, 1986.

\bibitem{1}
George~E. Andrews, {\em The {T}heory of {P}artitions}, Cambridge Mathematical
  Library, Cambridge University Press, 1998.

\bibitem{3}
George~E. Andrews, Concave compositions, {\em Electron. J. Combin.} {\bf 18} (2)
  (2011), Paper 6.

\bibitem{5}
George~E. Andrews and F.~G. Garvan, Dyson's crank of a partition, {\em Bull.
  Amer. Math. Soc.} {\bf 18} (1988), 167--171.

\bibitem{6}
Richard Blecksmith, John Brillhart, and Irving Gerst, Parity results for
  certain partition functions and identities similar to theta function
  identities, {\em Math. Comp.} {\bf 48} (1987), 29--38.

\bibitem{7}
F.~J. Dyson, Some guesses in the theory of partitions, {\em Eureka} 
{\bf 8} (1944), 10--15.

\bibitem{8}
Aviezri~S. Fraenkel and Udi Peled, Harnessing the unwieldy {MEX} function.
\newblock In {\em Games Of No Chance 4}, Vol.~63 of {\em Math. Sci. Res. Inst.
  Publ.}, pp.~77--94. Cambridge Univ. Press, 2015.

\bibitem{9}
Brian Hopkins and James~A. Sellers, Exact enumeration of {G}arden of {E}den
  partitions.
\newblock In B. Landman, M. Nathanson, J. Ne{\v s}et{\v r}il, R. Nowakowski and C. Pomerance, eds., {\em Combinatorial {N}umber {T}heory}, pp.~299--303. De Gruyter,
  Berlin, 2007.

\bibitem{10}
M.~Somos, contribution to sequence \seqnum{A064428}, in
N. J. A. Sloane et al.,
{\it On-Line Encyclopedia of Integer Sequences}.
\newblock Posted January 15 2018.

\bibitem{11}
Ali~Kemal Uncu, Weighted {R}ogers-{R}amanujan partitions and {D}yson crank,
  {\em Ramanujan J.} {\bf 46} (2018), 579--591.

\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11P81;
Secondary 11A63, 05A19.

\noindent \emph{Keywords: } 
minimal excludant, $\MEX$ function, theta function, partition.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A027187},
\seqnum{A046682},
\seqnum{A064428}, and
\seqnum{A260894}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received February 24 2019;
revised versions received  
July 24 2019; October 26 2019; October 30 2019; February 16 2020.
Published in {\it Journal of Integer Sequences}, February 17 2020.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in

\end{document}

                                                                                



