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%%command
\newcommand{\So}{\textbf{Step 1 }}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Soo}{\textbf{Step 1}}
\newcommand{\St}{\textbf{Step 2 }}
\newcommand{\Stt}{\textbf{Step 2}}
\newcommand{\Od}{\mathcal{O}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\Cc}{\mathcal{C}_\rtimes}
\newcommand{\D}{\mathcal{D}}
\newcommand{\Pp}{\mathcal{P}}
\newcommand{\Sc}{\mathcal{S}}
\newcommand{\Ga}{\Gamma}
\newcommand{\Gi}{\Gamma^{-1}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\Ee}{\mathcal{E}_1}
\newcommand{\Eee}{\mathcal{E}_2}
\newcommand{\ab}{\textcolor{blue}{a}}
\newcommand{\la}{\lambda}


%% define your title in the usual way
\title[Beyond G\"ollnitz' Theorem I]{Beyond G\"ollnitz' Theorem I: A Bijective Approach}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Isaac Konan]{Isaac Konan\thanks{\href{mailto:konan@irif.fr}{konan@irif.fr}.}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}IRIF, University of Paris Diderot, Paris, France}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{In 2003, Alladi, Andrews and Berkovich proved an identity for
partitions where parts occur in eleven colors: four primary colors,
six secondary colors, and one quaternary color.    Their work answered
a longstanding question of how to go beyond a classical theorem of
G\"ollnitz, which uses three primary and three secondary colors.
Their main tool was a deep and difficult four parameter $q$-series
identity. In this extended abstract, we take a different approach.   Instead of
adding an eleventh quaternary color, we introduce forbidden patterns
and give a bijective proof of a ten-colored partition identity lying
beyond G\"ollnitz' theorem.   Using a second bijection, we show that our
identity is equivalent to the identity of Alladi, Andrews, and
Berkovich. From a combinatorial viewpoint, the use of forbidden
patterns is more natural and leads to a simpler formulation.
In fact, in Part II following the full paper, we show how our method can be
used to go beyond G\"ollnitz' theorem to any number of primary colors.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Partitions, Rogers--Ramanujan type identities}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction and Statements of Results}
\subsection{History}

A partition of a positive integer $n$ is a non-increasing sequence of positive integers whose sum is equal to $n$. For example, the partitions of $7$ are 
\[(7),(6,1),(5,2),(5,1,1),(4,3),(4,2,1),(4,1,1,1),(3,3,1),(3,2,2),(3,2,1,1),\]
\[(3,1,1,1,1),(2,2,2,1),(2,2,1,1,1),(2,1,1,1,1,1)\,\,\text{and}\,\,(1,1,1,1,1,1,1)\,\cdot\] 
The study of partition identities has a long history, dating back to
Euler's proof that there are as many partitions of $n$ into distinct parts as partitions of $n$ into odd parts. The corresponding identity is 
\begin{equation}
(-q;q)_\infty = \frac{1}{(q;q^2)_\infty}\,,
\end{equation}
where $(x;q)_m = \prod_{k=0}^{m-1} (1-xq^k)$,
for  any $m\in \mathbb{N}\cup \{\infty\}$ and $x,q$ such that $|q|<1$.
%\\\\

One of the most important identities in the theory of partitions is
Schur's theorem \cite{Sc26}.
\begin{theorem}[Schur 1926]\label{scr}
For any positive integer $n$, the number of partitions of $n$ into distinct parts congruent to $\pm 1 \mod 3$ is equal to the number of partitions of $n$ where parts differ by at least three and multiples of three differ by at least six.
\end{theorem}
There have been a number of proofs of Schur's result over the years,
including a $q$-difference equation proof of Andrews \cite{AN68} and a simple
bijective proof of Bressoud \cite{BR80}.
%\\\\

Another important identity is G\"ollnitz' theorem \cite{GO67}.
\begin{theorem}[G\"ollnitz 1967]\label{goll}
For any positive integer $n$, the number of partitions of $n$ into distinct parts  congruent to $2,4,5 \mod 6$ is equal to the number of partitions of $n$ into parts  different from $1$ and $3$, and where parts differ by at least six with equality only if parts are congruent to $2,4,5 \mod 6$.
\end{theorem}
Like Schur's theorem, G\"ollnitz's identity can be proved using
$q$-difference equations \cite{AN69} and an elegant Bressoud-style bijection \cite{PRS04,ZJ15}.
%\\\\

Seminal work of Alladi, Andrews, and Gordon in the 90's showed how the
theorems of Schur and G\"ollnitz emerge from more general results on
colored partitions \cite{AAG95}. 
%\\\\

In the case of Schur's theorem, we
consider parts in three colors $\{a,b,ab\}$ and the partitions with colored parts different from $1_{ab}$ and satisfying the minimal difference conditions in the table 
\begin{equation}\label{scrrr}
\begin{array}{|c|cc|c|}
\hline
_{\lambda_i}\setminus^{\lambda_{i+1}}&a&b&ab\\
\hline
a&1&2&1\\
b&1&1&1\\
\hline
ab&2&2&2\\
\hline
\end{array}\,\cdot
\end{equation}
Here, the part $\la_i$ with color in the row and the part $\la_{i+1}$ with color in the column differ by at least the corresponding entry in the table.  
An example of such a partition is $(7_{ab},5_{b},4_{a},3_{ab},1_b)$. The Alladi-Gordon refinement of Schur's partition theorem \cite{AG93} is stated as follows:
\begin{theorem}\label{ag}
Let $u,v,n$ be non-negative integers. Denote by $A(u,v,n)$ the number of partitions of $n$ into $u$ distncts parts with color $a$ and $v$ distinct parts with color $b$, and denote by $B(u,v,n)$ the number of partitions of $n$ satisfying the conditions above, with $u$ parts with color $a$ or $ab$, and $v$ parts with color $b$ or $ab$. We then have  $A(u,v,n)=B(u,v,n)$ and the identity
\begin{equation}
\sum_{u,v,n\geq 0} B(u,v,n)a^ub^vq^n = \sum_{u,v,n\geq 0} A(u,v,n)a^ub^vq^n = (-aq;q)_\infty (-bq;q)_\infty\,\cdot
\end{equation}
\end{theorem}
Note that a transformation implies Schur's theorem :
\begin{equation}
\left\lbrace 
\begin{array}{l r c l }
\text{dilation :} &q &\mapsto&q^3\\
\text{translations :} &a,b &\mapsto&q^{-2},q^{-1}\\
\end{array}
\right. \,\cdot
\end{equation}
In fact, the minimal difference conditions given in \eqref{scrrr} give after these transformations the minimal differences in Schur's theorem. 
%\\\\

In the case of G\"ollnitz' theorem,
we consider parts that occur in six colors $\{a,b,c,\linebreak[0]ab,\linebreak[0]ab,bc\}$ 
and the partitions with colored parts different from $1_{ab},1_{ac},1_{bc}$ and satisfying the minimal difference conditions in 
\begin{equation}
\begin{array}{|c|ccc|ccc|}
\hline
_{\lambda_i}\setminus^{\lambda_{i+1}}&a&b&c&ab&ac&bc\\
\hline
a&1&2&2&1&1&2\\
b&1&1&2&1&1&1\\
c&1&1&1&1&1&1\\
\hline
ab&2&2&2&2&2&2\\
ac&2&2&2&1&2&2\\
bc&1&2&2&1&1&2\\
\hline
\end{array}\,\cdot
\end{equation}
The Alladi--Andrews--Gordon refinement of G\"ollnitz's partition theorem can be stated as follows:
\begin{theorem}\label{aag}
Let $u,v,w,n$ be non-negative integers. Denote by $A(u,v,w,n)$ the number of partitions of $n$ into $u$ distncts parts with color $a$, $v$ distinct parts with color $b$ and $w$ distinct parts with color $c$, and denote by $B(u,v,w,n)$ the number of partitions of $n$ satisfying the conditions above, with $u$ parts with color $a,ab$ or $ac$, $v$ parts with color $b,ab$ or $bc$ and $w$ parts with color $c,ac$ or $bc$. We then have  $A(u,v,w,n)=B(u,v,w,n)$ and the identity
\begin{small}
\begin{equation}
\sum_{u,v,w,n\geq 0} B(u,v,w,n)a^ub^vc^wq^n = \sum_{u,v,w,n\geq 0} A(u,v,w,n)a^ub^vc^wq^n = (-aq;q)_\infty (-bq;q)_\infty(-cq;q)_\infty\,\cdot
\end{equation}
\end{small}
\end{theorem}
Note that a transformation implies G\"ollnitz' theorem :
\begin{equation}
\left\lbrace 
\begin{array}{l r c l }
\text{dilation :} &q &\mapsto&q^6\\
\text{translations :} &a,b,c &\mapsto&q^{-4},q^{-2},q^{-1}\\
\end{array}
\right. \,\cdot
\end{equation}
%\\
Observe that while Schur's theorem is not a direct corollary of
G\"ollnitz' theorem, \cref{ag} \emph{is} implied by \cref{aag} by
setting $c= 0$.    Therefore G\"ollnitz' theorem may be viewed as a level
higher than Schur's theorem, since it requires three primary colors
instead of two.
%\\\\

Following the work of Alladi, Andrews, and Gordon, it was an open
problem to find a partition identity beyond G\"ollnitz' theorem, in the
sense that it would arise from four primary colors.    This was
famously solved by Alladi, Andrews, and Berkovich \cite{AAB03}. To describe their result, 
we consider parts that occur in eleven colors
$\{a,b,c,d,\linebreak[0]ab,\linebreak[0]ab,\linebreak[0]ad,\linebreak[0]bc,bd,cd,abcd\}$.
We now take the partitions with the length of the secondary parts greater than one and satisfying the minimal difference conditions in 
\begin{equation}\label{tab2}
\begin{array}{|c|cccc|ccc|cc|c|}
\hline
_{\lambda_i}\setminus^{\lambda_{i+1}}&ab&ac&ad&a&bc&bd&b&cd&c&d\\
\hline
ab&2&2&2&2&2&2&2&2&2&2\\
ac&1&2&2&2&2&2&2&2&2&2\\
ad&1&1&2&2&2&2&2&2&2&2\\
a&1&1&1&1&2&2&2&2&2&2\\
\hline
bc&1&1&1&1&2&2&2&2&2&2\\
bd&1&1&1&1&1&2&2&2&2&2\\
b&1&1&1&1&1&1&1&2&2&2\\
\hline
cd&1&1&1&1&1&1&1&2&2&2\\
c&1&1&1&1&1&1&1&1&1&2\\
\hline
d&1&1&1&1&1&1&1&1&1&1\\
\hline
\end{array}\,,
\end{equation}
and such that parts with color $abcd$ differ by at least $4$, and the smallest part with color $abcd$ is at least equal to  $4+2\tau-\chi(1_a\text{ is a part})$, where $\tau$ is the number of primary and secondary parts in the partition.
The theorem of Alladi, Andrews, and Berkovich is then stated as follows.
\begin{theorem}\label{th2}
Let $u,v,w,t,n$ be non-negative integers. Denote by $A(u,v,w,t,n)$ the number of partitions of $n$ into $u$ distncts parts with color $a$, $v$ distinct parts with color $b$, $w$ distinct parts with color $c$ and $t$ distinct parts with color $d$, and denote by $B(u,v,w,t,n)$ the number of partitions of $n$ satisfying the conditions above, with $u$ parts with color $a,ab,ac,ad$ or $abcd$, $v$ parts with color $b,ab,bc,bd$ or $abcd$, $w$ parts with color $c,ac,bc,cd$ or $abcd$ and $t$ parts with color $d,ad,bd,cd$ or $abcd$.
We then have $A(u,v,w,t,n)=B(u,v,w,t,n)$ and the identity
\begin{equation}
\sum_{u,v,w,t,n\geq 0} B(u,v,w,t,n)a^ub^vc^wd^tq^n =
(-aq;q)_\infty (-bq;q)_\infty(-cq;q)_\infty(-dq;q)_\infty\,\cdot
\end{equation}
\end{theorem}
Note that the result of Alladi--Andrews--Berkovich  uses four primary colors, the full set of
secondary colors, along with one quaternary color $abcd$. When $d=0$, 
we recover \cref{aag}. Their main tool was a difficult $q$-series
identity.
%\\\\

In this extended abstract (for the full paper, see \cite{IK19}), we present a bijective proof of \cref{th2}.   Our
proof is divided into two steps.  First we prove \cref{th1} below,
which arises more naturally from our methods than \cref{th2}.
Instead of adding a quaternary color, we lower certain minimum
differences and add some forbidden patterns.  Then, we show how
\cref{th1} is equivalent to \cref{th2}.
%\\\\

The general result beyond G\"ollnitz's theorem for an arbitrary number of primary colors is given in paper two of this series \cite{IK192}. 

\subsection{Statement of Results}
Suppose that the parts occur in only primary colors $a,b,c,d$ and secondary colors\linebreak 
$ab,ac,ad,bc,bd,cd$. Let us now consider the partitions with the length of the secondary parts greater than one and satisfying the minimal difference conditions in \eqref{tab2}, where we allow the patterns
\begin{equation}
(k_{cd},k_{ab}), ((k+1)_{ad},k_{bc})\,,
\end{equation}
while avoiding the following forbidden patterns for $k\geq 3$ :
\begin{equation}
(k_{cd},k_{ab},(k-2)_c),(k_{cd},k_{ab},(k-2)_d), ((k+1)_{ad},k_{bc},(k-1)_{a})\,\cdot
\end{equation}
An example of such a partition is 
\[(11_{ad},10_{bc},8_a,7_{cd},7_{ab},4_c,3_{ad},2_{bc},1_a)\,\cdot\]
We can now state the main theorem of this paper.
\begin{theorem}\label{th1}
Let $u,v,w,t,n$ be non-negative integers. Denote by $A(u,v,w,t,n)$ the number of partitions of $n$ into $u$ distinct parts with color $a$, $v$ distinct parts with color $b$, $w$ distinct parts with color $c$ and $t$ distinct parts with color $d$, and denote by $B(u,v,w,t,n)$ the number of partitions of $n$ satisfying the conditions above, with $u$ parts with color $a,ab,ac$ or $ad$, $v$ parts with color $b,ab,bc$ or $bd$, $w$ parts with color $c,ac,bc$ or $cd$ and $t$ parts with color $d,ad,bd$ or $cd$.
We then have $A(u,v,w,t,n)=B(u,v,w,t,n)$, and the corresponding $q$-series identity is given by
\begin{equation}\label{series}
\sum_{u,v,w,t,n\in \N} B(u,v,w,t,n)a^ub^vc^wd^tq^n =(-aq;q)_\infty (-bq;q)_\infty(-cq;q)_\infty(-dq;q)_\infty\,\cdot
\end{equation}
\end{theorem}
By
specializing the variables in \cref{th1}, one can deduce
many partition identities.   For example, by considering the following transformation in \eqref{series}
\begin{equation}\label{dila}
\left\lbrace 
\begin{array}{l r c l }
\text{dilation :} &q &\mapsto&q^{12}\\
\text{translations :} &a,b,c,d &\mapsto&q^{-8},q^{-4},q^{-2},q^{-1}\\
\end{array}
\right. \,,
\end{equation}
we obtain a corollary of \cref{th1}.
\begin{corollary}
For any positive integer $n$, the number of partitions of $n$ into distinct parts congruent to $-2^3,-2^2,-2^1,-2^0\mod 12$ is equal to the number of partitions of $n$ into parts not congruent to $1,5\mod 12$ and different from $2,3,6,7,9$, such that the difference between two consecutive parts is greater than $12$ up to the following exceptions:
\begin{itemize}
\item $\la_i-\la_{i+1}= 9\Longrightarrow \la_i\equiv \pm 3 \mod 12$ and $\la_i-\la_{i+2}\geq 24$,
\item $\la_i-\la_{i+1}= 12\Longrightarrow\la_i\equiv -2^3,-2^2,-2^1,-2^0\mod 12$,
\end{itemize}
except that the pattern $(27,18,4)$ is allowed. 
\end{corollary}
For example, with $n=49$, the partitions of the first kind are 
\[(35,10,4),(34,11,4),(28,11,10),(23,22,4),\]
\[(23,16,10),(22,16,11)\,\,\text{and}\,\, (16,11,10,8,4)\]
and the partitions of the second kind are
\[(35,14),(34,15),(33,16),(45,4),(39,10),(38,11)\,\,\text{and}\,\,(27,18,4)\,\cdot\]
\textbf{Corollary 1.1} may be compared
with \textbf{Theorem 3} of \cite{AAB03}, which is \cref{th2} transformed by \eqref{dila} but
with the dilation $q \mapsto q^{15}$ instead of $q \mapsto q^{12}$.
%\\\\

The extended abstract is organized as follows. In \cref{sct2}, we will present some tools that will be useful for the bijections. 
After that, in \cref{sct3}, we will give the bijection for \cref{th1}. 
Finally, in \cref{sct5}, we will present the bijection between the partitions with forbidden patterns considered in \cref{th1} and the partitions with quaternary parts given in \cref{th2}.
\section{The setup}\label{sct2}
Denote by $\C=\{a,b,c,d\}$ the set of primary colors  and $\Cc=\{ad,ab,ac,bc,bd,cd\}$ the set of secondary colors, and recall the order on $\C\sqcup\Cc$: 
\begin{equation}\label{orD}
ab<ac<ad<a<bc<bd<b<cd<c<d\,\cdot
\end{equation}
We can then define the strict lexicographic order $\succ$ on colored parts by
\begin{equation}\label{lex}
k_p\succ l_{q} \Longleftrightarrow k-l\geq \chi(p\leq q)\,\cdot
\end{equation}
Explicitly, this gives the order 
\begin{equation}\label{cons1}
1_{ab}\prec 1_{ac}\prec 1_{ad} \prec 1_{a}\prec 1_{bc} \prec 1_{bd}\prec 1_b\prec 1_{cd}\prec 1_c\prec 1_d\prec 2_{ab}\prec\cdots\,\cdot
\end{equation}
We denote by $\Pp$ the set of positive integers with primary color. We can easily see that for any $pq\in \Cc$, \textit{with} $p<q$, and any $k\geq 1$, we have that 
\begin{align}
\label{half1}(2k)_{pq} &= k_{q}+k_{p}\\
\label{half2}(2k+1)_{pq} &= (k+1)_{p}+k_{q}\,\cdot
\end{align}
Here, the sum of two parts with primary colors consists of a part whose size and color are respectively the sum  of the sizes and the commutative product of colors of the primary colored parts. 
%\\
One can check that any part greater than $1$ with a secondary color $pq$ can be uniquely written as the sum of two consecutive parts in $\Pp$ with colors $p$ and $q$. We then denote by $\Sc$ the set of secondary parts greater than $1$, and define the functions $\alpha$ and $\beta$ on $\Sc$ by 
\begin{equation}\label{ab}
\alpha: \left\lbrace \begin{array}{l c l}
2k_{pq}&\mapsto& k_{q}\\
(2k+1)_{pq}&\mapsto&(k+1)_{p}
\end{array}\right.\qquad \text{and}\qquad\beta: \left\lbrace \begin{array}{l c l}
2k_{pq}&\mapsto& k_{p}\\
(2k+1)_{pq}&\mapsto&k_{q}
\end{array}\right.\,,
\end{equation}
respectively named upper and lower halves.
%\\\\

By considering the lexicographic order $\succ$, the minimal differences described in the table \eqref{tab2} can be viewed as an order $\triangleright$ on $\Pp\sqcup \Sc$ 
defined by the relation
\begin{equation}\label{Ord}
k_p\triangleright l_{q}\Longleftrightarrow\left\lbrace
\begin{array}{ll}
k_p\succeq (l+1)_{q}&\text{if}\quad p\,\,\text{or}\,\,q\in \C\\
k_p\succ (l+1)_{q}&\text{if}\quad p\,\,\text{and}\,\,q\in \Cc
\end{array}\right.\,\cdot
\end{equation}
We recall that the table \eqref{tab2} 
and the minimal differences for \cref{th1} differ only when we have a pair $(p,q)$ of secondary colors such that $(p,q)\in \{(cd,ab),(ad,bc)\}$.  In these cases, the difference for \cref{th1} is one less.
We then define in the same way a relation  $\gg$ on $\Pp\sqcup \Sc$, for the minimal differences of \cref{th1}, and obtain by \eqref{Ord}
\begin{equation}\label{Ordd}
k_p\gg l_{q}\Longleftrightarrow\left\lbrace
\begin{array}{ll}
k_p\succeq (l+1)_{q}&\text{if}\quad p\,\,\text{or}\,\,q \in \C\\
k_p\succ (l+1)_{q}&\text{if}\quad p\,\,\text{and}\,\,q \in \Cc\quad\text{and}\quad (p,q)\notin\{(cd,ab),(ad,bc)\}\\
k_p\succ l_{q}&\text{if}\quad (p,q)\in\{(cd,ab),(ad,bc)\}
\end{array}\right.\,\cdot
\end{equation}
We denote by $\Od$ the set of partitions with parts in $\Pp$ and well-ordered by $\succ$. We then have that 
$\la \in \Od$ if and only if there exist $\la_1\succ\cdots\succ \la_t \in \Pp$ such that $\la=(\la_1,\ldots,\la_t)$. We set 
$c(\la_i)$ to be the color of $\la_i$ in $\C$, and $C(\la) = c(\la_1)\cdots c(\la_t)$ as a commutative product of colors in $<\C>$.
In the same way, we denote by $\E$ the set of partitions with parts in $\Pp\sqcup \Sc$ and well-ordered by $\gg$ and set colors $c(\nu_i)\in \C\sqcup\Cc$ depending on whether $\nu_i$ is in  $\Pp$ or $\Sc$, 
and we also define 
$C(\nu)=c(\nu_1)\cdots c(\nu_t)$ \textit{seen} as a commutative product of colors in $\C$. In fact, a secondary color is just a product of two primary colors. For both kinds of partitions, their size is the sum of their part sizes. 
We also denote by $\Ee$ the subset of partitions of $\E$ with the forbidden patterns,
\begin{equation}\label{forb}
((k+2)_{cd},(k+2)_{ab},k_c),((k+2)_{cd},(k+2)_{ab},k_d), ((k+2)_{ad},(k+1)_{bc},k_{a})\,,
\end{equation}
except the pattern $(3_{ad},2_{bc},1_a)$ which is allowed. 
We finally define $\Eee$ as the subset of partitions of $\E$ with parts well-ordered by $\triangleright$ in \eqref{Ord}, and we observe that $\Eee$ is indeed a subset of $\Ee$. 
%\\\\

Finally, in what follows, adding an integer to a colored part only changes its size and does not affect its color.
\section{Bressoud's algorithm}\label{sct3}
Here we adapt the algorithm given by Bressoud in his bijective proof of Schur's partition theorem \cite{BR80}. The bijection is easy to
describe and execute, but its justification is more subtle and is
given in Section $4$ of the full paper \cite{IK19}. 
\subsection{From $\Od$ to $\Ee$} Let us consider the following machine $\Phi$:
\setlist{leftmargin=0.6in}
\begin{itemize}
\item[\Soo:]For a sequence $\la= \la_1,\ldots,\la_t$, take the smallest $i<t$ such that  $\la_i,\la_{i+1}\in \Pp$ and $\la_i\succ \la_{i+1}$ but $\la_i\not\gg \la_{i+1}$, if it exists, and replace
\begin{equation}
\begin{array}{l c l l}
\la_i &\leftarrowtail& \la_i+\la_{i+1}&\text{as a part in } \Sc\\
\la_j &\leftarrow& \la_{j+1}& \text{for all}\quad i<j<t\,\, 
\end{array}
\end{equation}
and move to \Stt. We call such a pair of parts a \textit{troublesome} pair. We observe that $\la$ loses two parts in $\Pp$ and gains one part in $\Sc$. 
The new sequence is $\la = \la_1,\ldots,\la_{t-1}$. Otherwise, exit from the machine.%\\
\item[\Stt:]For $\la= \la_1,\ldots,\la_t$, take the smallest $i<t$ such that $(\la_i,\la_{i+1})\in \Pp\times\Sc$ and $\la_i\not\gg \la_{i+1}$ if it exists,  and replace 
\begin{equation}
(\la_i,\la_{i+1}) \looparrowright (\la_{i+1}+1,\la_i-1)\in \Sc\times\Pp
\end{equation}
and redo \Stt. We say that the parts $\la_i,\la_{i+1}$ are \textit{crossed}. Otherwise, move to \Soo.
\end{itemize} 
Let $\Phi(\la)$ be the resulting sequence after putting  any $\la=(\la_1,\ldots,\la_t)\in \Od$ in $\Phi$. 
This transformation preserves the size and the commutative product of primary colors of partitions.  Below, we apply this machine on the partition \[
(11_c,8_d,6_a,4_d,4_c,4_b,3_a,2_b,2_a,1_d,1_c,1_b,1_a).
\]
\begin{footnotesize}
\[ \begin{array}{ccccccccccccccccc}
\begin{matrix}
11_c\\
8_d\\
6_a\\
\underline{4_d}\\
\underline{4_c}\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix} &\rightarrowtail&
\begin{matrix}
11_c\\
8_d\\
\mathbf{6_a}\\
\mathbf{8_{cd}}\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}&
\looparrowright&
\begin{matrix}
11_c\\
\mathbf{8_d}\\
\mathbf{9_{cd}}\\
5_a\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\looparrowright&
\begin{matrix}
11_c\\
10_{cd}\\
7_d\\
\underline{5_a}\\
\underline{4_b}\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\rightarrowtail&
\begin{matrix}
11_c\\
10_{cd}\\
\mathbf{7_d}\\
\mathbf{9_{ab}}\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\looparrowright&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
\underline{3_a}\\
\underline{2_b}\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\rightarrowtail&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
5_{ab}\\
\underline{2_a}\\
\underline{1_d}\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\rightarrowtail&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
5_{ab}\\
3_{ad}\\
\underline{1_c}\\
\underline{1_b}\\
1_a
\end{matrix}
&
\rightarrowtail&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
5_{ab}\\
3_{ad}\\
2_{bc}\\
1_a
\end{matrix}
\end{array}\,\cdot
\]
\end{footnotesize}
\subsection{From $\Ee$ to $\Od$}
Let us consider the following machine $\Psi$:
\begin{itemize}
\item[\Soo:]For a sequence $\nu= \nu_1,\ldots,\nu_t$, take the greastest $i\leq t$ such that $\nu_i\in \Sc$ if it exists.  If $\nu_{i+1}\in\Pp$ and $\beta(\nu_i)\not\succ \nu_{i+1}$, then replace 
\begin{equation}
(\nu_i,\nu_{i+1}) \looparrowright (\nu_{i+1}+1,\nu_i-1)\in \Pp\times\Sc
\end{equation}
and redo \Soo. We say that the parts $\nu_i,\nu_{i+1}$ are \textit{crossed}. Otherwise, move to \Stt. If there are no more parts in $\Sc$, exit from the machine.%\\
\item[\Stt:]For $\nu= \nu_1,\ldots,\nu_t$, take the the greatest $i\leq t$ such that $\nu_i\in \Sc$. By \Soo, it satisfies $\beta(\nu_i)\succ \nu_{i+1}$. Then replace
\begin{equation}
\begin{array}{l c l l}
\nu_{j+1} &\leftarrow& \nu_{j}& \text{for all}\quad t\geq j>i\,\, \\
(\nu_i) &\rightrightarrows& (\alpha(\nu_i),\beta(\nu_i))&\text{as a pair of parts in }\Pp \,,
\end{array}
\end{equation}
and move to \Soo.  We say that the part $\nu_i$ \textit{splits}. We observe that $\nu$ gains two parts in $\Pp$ and loses one part in $\Sc$. 
The new sequence  is $\nu = \nu_1,\ldots,\nu_{t+1}$.\\
\end{itemize}
Let $\Psi(\nu)$ be the resulting sequence after putting  any $\nu=(\nu_1,\ldots,\nu_t)\in \Ee$ in $\Psi$.
This transformation preserves the size and the product of primary colors of partitions. For example, applying this to $(11_c,10_{cd},10_{ab},6_d,5_{ab},3_{ad},2_{bc},1_a)$ gives
\begin{tiny}
\[ \begin{array}{ccccccccccccccccc}
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
5_{ab}\\
3_{ad}\\
1_c+1_b\\
1_a
\end{matrix} &
\rightrightarrows&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
5_{ab}\\
2_a+1_d\\
1_c\\
1_b\\
1_a
\end{matrix} &
\rightrightarrows&
\begin{matrix}
11_c\\
10_{cd}\\
10_{ab}\\
6_d\\
3_a+2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}&
\rightrightarrows&
\begin{matrix}
11_c\\
10_{cd}\\
5_b+5_a\\
6_d\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\looparrowright&
\begin{matrix}
11_c\\
10_{cd}\\
7_d\\
5_a+4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\rightrightarrows&
\begin{matrix}
11_c\\
5_d+5_c\\
7_d\\
5_a\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\looparrowright&
\begin{matrix}
11_c\\
8_d\\
5_c+4_d\\
5_a\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
&
\looparrowright&
\begin{matrix}
11_c\\
8_d\\
6_a\\
4_d+4_c\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}&
\rightrightarrows&
\begin{matrix}
11_c\\
8_d\\
6_a\\
4_d\\
4_c\\
4_b\\
3_a\\
2_b\\
2_a\\
1_d\\
1_c\\
1_b\\
1_a
\end{matrix}
\end{array}\,\cdot
\]  
\end{tiny}
\section{Bijective proof of \cref{th2}}\label{sct5}

In this section, we will describe a bijection for \cref{th2}. For brevity, we refer to the partitions in \cref{th2} as quaternary partitions.
\subsection{From $\Ee$ to quaternary partitions}

We consider the patterns $((k+1)_{ad},k_{bc}),(k_{cd},k_{ab})$ and we sum them as follows :
\begin{align}
(k+1)_{ad}+k_{bc}&= (2k+1)_{abcd}\nonumber\\
k_{cd}+k_{ab}&= 2k_{abcd}\,\label{decomp}\,\cdot
\end{align}
Let us now take a partition $\nu$ in $\Ee$.
We then identify all the patterns $(M^i,m^i)\in \{((k+1)_{ad},k_{bc}),(k_{cd},k_{ab})\}$ and suppose that 
\[\nu = \nu_1,\ldots,\nu_x,M^1,m^1,\nu_{x+1},\ldots,\nu_y,M^2,m^2,\nu_{y+1},\ldots, M^t,m^t,\ldots,\nu_s\,\cdot\]
\textit{As long as} we have a pattern $\nu_j,M^i,m^i$, we cross the parts by replacing them using 
\begin{equation}
\nu_j,M^i,m^i \longmapsto M^i+1,m^i+1,\nu_j-2\,\cdot
\end{equation}
At the end of the process, we obtain a final sequence 
\[N^1,n^1,N^2,n^2,\ldots,N^t,n^t,\nu'_1,\ldots,\nu'_s\,\cdot\]
Finally, the associated pair of partitions is set to be $(K^1,\ldots,K^t),\nu'=(\nu'_1,\ldots,\nu'_t)$, where 
$K^i=N^i+n^i$ according to  \eqref{decomp}. To sum up the previous transformation, we only remark that, for each quaternary part $K^i$ obtained by summing of the original pattern $M^i,m^i$, we add twice the number of the remaining primary and secondary parts in $\nu$ to the left of the pattern that gave $K^i$, 
while we subtract from these parts two times the number of quaternary parts obtained by patterns that occur to their right.
%\\\\

With the example $11_c,\underline{10_{cd},10_{ab}},6_d,5_{ab},\underline{3_{ad},2_{bc}},1_a$,
\[
\begin{footnotesize}
\begin{array}{ccccccccccccc}\begin{matrix}
 11_c\\
 10_{cd}\\
 10_{ab}\\
 6_d\\
 5_{ab}\\
 3_{ad}\\
  2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_c\\
 10_{cd},10_{ab}\\
 6_d\\
 5_{ab}\\
 3_{ad}, 2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 6_d\\
 5_{ab}\\
 3_{ad},2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 6_d\\
 4_{ad},3_{bc}\\
 3_{ab}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 5_{ad},4_{bc}\\
 4_d\\
 3_{ab}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
6_{ad},5_{bc}\\
 7_c\\
 4_d\\
 3_{ab}\\
 1_a
\end{matrix}
\end{array}\,\cdot
\end{footnotesize}
\]
 we obtain $[(22_{abcd},11_{abcd}),(7_c,4_d,3_{ab},1_a)]$.
\subsection{From quaternary partitions to $\Ee$}
Recall  by \eqref{decomp} that $K_{abcd}$ splits as follows :
\begin{align*}
(k+1)_{ad}+k_{bc}&= (2k+1)_{abcd}\\
k_{cd}+k_{ab}&= 2k_{abcd}\,
\end{align*}
Let  us then consider partitions 
$(K^1,\ldots,K^t)$ and $\nu=(\nu_1,\ldots,\nu_s)\in \Eee$, with quaternary part $K^u$ such that $K^t\geq 4+2s-\chi(1_a\in \nu)$ and $K^u-K^{u+1}\geq 4$.
We also set $K^u = (k^u,l^u)$ the decomposotion according to \eqref{decomp}. We then proceed as follows by beginning with $K^t$ and $\nu_1$, 
\begin{itemize}
\item[\Soo:] If we do not encounter $K^{u+1}=(k^{u+1},l^{u+1})$ and $\nu_i\neq 1_a$ and $\nu_i+2\triangleright k^u-1$, then replace
\begin{align*}
\nu_i&\longmapsto \nu_i+2\\
(k^u,l^u)&\longmapsto (k^u-1,l^u-1)
\end{align*}
and move to $i+1$ and redo \Soo. Otherwise, move to \Stt.
\item[\Stt] If we encounter $K^{u+1}= k^{u+1}\gg l^{u+1}$, then split $(k^u,l^u)$ into $k^u\gg l^u$.
If not, it means that we have met $\nu_i$ such that $\nu_i+2\not\triangleright\,\,\, k^u-1$. Then we split $k^u\gg l^u$.
We can now move to \So with $u-1$ and $i=1$.
\end{itemize}
With the example $[(22_{abcd},11_{abcd}),(7_c,4_d,3_{ab},1_a)]$, we obtain
\[
\begin{scriptsize}
\begin{array}{ccccccccccccccccc}
\begin{matrix}
 11_{cd},11_{ab}\\
6_{ad},5_{bc}\\
 7_c\\
 4_d\\
 3_{ab}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 5_{ad},4_{bc}\\
 4_d\\
 3_{ab}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 6_d\\
 4_{ad},3_{bc}\\
 3_{ab}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 6_d\\
 5_{ab}\\
 3_{ad},2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_{cd},11_{ab}\\
 9_c\\
 6_d\\
 5_{ab}\\
 3_{ad}\\
 2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_c\\
 10_{cd},10_{ab}\\
 6_d\\
 5_{ab}\\
 3_{ad}\\
 2_{bc}\\
 1_a
\end{matrix}
&\mapsto&
\begin{matrix}
 11_c\\
 10_{cd}\\
 10_{ab}\\
 6_d\\
 5_{ab}\\
 3_{ad}\\
 2_{bc}\\
 1_a
\end{matrix}
\end{array}\,\cdot
\end{scriptsize}
\]

\acknowledgements{Thanks to Jeremy Lovejoy for suggesting the author work on G\"ollnitz' theorem.}

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\end{document}
