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\begin{center}
\vskip 1cm{\LARGE\bf 
Extending a Recent Result on \\ 
\vskip .1in
Hyper $m$-ary Partition Sequences
}
\vskip 1cm
\large
Timothy B. Flowers\\
Department of Mathematics\\
Indiana University of Pennsylvania\\
Indiana, PA 15705\\
USA\\
\href{mailto:flowers@iup.edu}{\tt flowers@iup.edu} \\
\ \\
Shannon R. Lockard\\
Department of Mathematics\\
Bridgewater State University\\
Bridgewater, MA 02324\\
USA\\
\href{mailto:Shannon.Lockard@bridgew.edu}{\tt Shannon.Lockard@bridgew.edu} \\
\end{center}

\vskip .2in

\begin{abstract}
A hyper $m$-ary partition of an integer $n$ is defined to be a partition of $n$ where each part is a power of $m$ and each distinct power of $m$ occurs at most $m$ times. Let $h_m(n)$ denote the number of hyper $m$-ary partitions of $n$ and consider the resulting sequence. We show that the hyper $m_1$-ary partition sequence is a subsequence of the hyper $m_2$-ary partition sequence, for $2 \leq m_1 < m_2$. 
\end{abstract}


\section{Introduction}
In 2004, Courtright and Sellers \cite{m-ary} defined a hyper $m$-ary partition of an integer $n$ to be a partition of $n$ for which each part is a power of $m$ and each power of $m$ occurs at most $m$ times.  They denote the number of hyper $m$-ary partitions of $n$ as $h_m(n)$ and showed that they satisfy the following recurrence relation:
\begin{eqnarray}
    h_m(mq) &=& h_m(q) + h_m(q-1),  \label{recur1} \\
    h_m(mq+s) &=& h_m(q), \text{ for } 1 \le s \leq m-1. \label{recur2}
\end{eqnarray}
Several of these hyper $m$-ary partition sequences can be found in the On-line Encyclopedia of Integer Sequences \cite{OEIS}.  In particular, $h_2$ is \seqnum{A002487}, $h_3$ is \seqnum{A054390}, $h_4$ is \seqnum{A277872}, and $h_5$ is \seqnum{A277873}.

The sequence $h_2$ \seqnum{A002487}, the hyperbinary partition sequence, is well known. It is commonly known as the Stern sequence based on Stern's work \cite{stern}. Northshield \cite{northshield} gives an extensive summary of the many uses and applications of \seqnum{A002487}. Calkin and Wilf \cite{recounting} also studied $h_2$, outlining a connection between this sequence and a sequence of fractions they defined and used to give an enumeration of the rationals. Since then, several authors have studied similar restricted binary and $m$-ary partition functions; see \cite{sohn, ours, reznick} for additional examples.

In this paper, we will be analyzing hyper $m$-ary partitions of $n$ while also considering the base $m$ representation of $n$. Thus, it will be convenient to have clear and distinct notation. In particular, for $m\geq 2$, let $(n_r,n_{r-1},\ldots,n_1,n_0)_m$ be the base $m$ representation of positive integer $n$ where $0 \leq n_i < m$, $n_r \neq 0$, and $n = \sum_{i=0}^{r} n_i m^i$. Also, for ${2 \leq m_1 < m_2}$ and $n = (n_r,n_{r-1}, \ldots,n_1,n_0)_{m_1}$, we define a change of base function, $F_{m_1,m_2}(n) = (n_r,n_{r-1}, \ldots,n_1,n_0)_{m_2}$.    

Next, we write a hyper $m$-ary partition of $n$ as $[x_r,x_{r-1}, \ldots,x_1,x_0]_m$ where $0 \leq x_i \leq m$ and $n = \sum_{i=0}^{r} x_i m^i$. Here, we may allow any of the $x_i$ to be 0 so that each hyper $m$-ary partition of $n$ is the same length $r$ as the base $m$ representation of $n$. Furthermore, let $H_m(n)$ be the set of all distinct hyper $m$-ary partitions of $n$. Observe that $h_m(n)$ is the cardinality of this set.

Recently, the authors gave an identity relating $h_2$ to $h_3$ and then generalized this identity to show that $h_2$ is a subsequence of $h_m$ for any $m$ \cite{ours}.  This result involved giving a bijection between $H_2(\ell)$ and $H_m(k)$, where $k = F_{2,m}(\ell)$. In this note, the authors will follow a similar process to show that $h_{m_1}$ is a subsequence of $h_{m_2}$, for $2 \leq m_1 < m_2$.


\section{A preliminary example}

Consider the integer $37 = (1,1,0,1)_3$ and use the change of base function to find the integer with the same digits in base 4.  In particular, $F_{3,4}(37) = (1,1,0,1)_4 = 81$. Now consider the hyper $3$-ary partitions of 37 and the hyper $4$-ary partitions of 81.

\begin{minipage}{.45\textwidth}
	\begin{eqnarray*}
 37 &=& 1 \cdot 3^3 + 1 \cdot 3^2 + 1 \cdot 3^0 \\
  	&=& 1 \cdot 3^3 + 3\cdot 3^1 + 1 \cdot 3^0 \\
  	&=& 3 \cdot 3^2 + 3\cdot 3^1 + 1\cdot 3^0
\end{eqnarray*}
\end{minipage}
\hfill
\begin{minipage}{.45\textwidth}
	\begin{eqnarray*}
  81 &=& 1 \cdot 4^3 + 1 \cdot 4^2 + 1 \cdot 4^0 \\
  	&=& 1 \cdot 4^3 + 4\cdot 4^1 + 1 \cdot 4^0 \\
  	&=& 4 \cdot 4^2 + 4\cdot 4^1 + 1\cdot 4^0
\end{eqnarray*}
\end{minipage}
 \\ \\
 
\noindent Adopting the notation for hyper $m$-ary partitions and the sets of these partitions,  rewrite these partitions in the following way:
\begin{eqnarray*}
H_3(37) &=& \left\{\, [1,1,0,1]_3,\,  [1,0,3,1]_3,\,  [0,3,3,1]_3 \,  \right\}; \\
H_4(81) &=& \left\{\, [1,1,0,1]_4,\,  [1,0,4,1]_4,\,  [0,4,4,1]_4 \,  \right\}.
\end{eqnarray*}
 
 Note that the number of hyper $3$-ary partitions of 37 is the same as the number of hyper $4$-ary partitions of 81. In other words, 
 $$h_3(37) = h_4\left(F_{3,4}(37)\right) = h_4(81) \, .$$ 
 We also observe that the coefficients of the partitions are similar, indicating that there is a relationship between the partitions in each set. This relationship will be further explored in the next section.

\section{Bijections between hyper $m$-ary partitions and hyper $(m+1)$-ary partitions}

We now verify the result suggested by the example in the prior section by considering hyper $m$-ary partitions of an integer $\ell$ and the hyper $(m+1)$-ary partitions of $k = F_{m,m+1}(\ell)$.

\begin{lemma}\label{mtom+1_bijection}
For a positive integer $\ell$, let $k = F_{m,m+1}(\ell)$. Define $g_m: H_{m+1}(k) \rightarrow H_m(\ell) $ by mapping
$$[c_r, c_{r-1}, \ldots, c_2 ,c_1 ,c_0]_{m+1} \mapsto [b_r ,b_{r-1}, \cdots, b_2, b_1, b_0]_{m}$$
according to the following rules:
\begin{align*}
c_i  = 0 &\longrightarrow  b_i = 0  \\ 
c_i = 1  &\longrightarrow b_i = 1  \\ 
& \, \, \, \,\,  \vdots \\
c_i = m-2 &\longrightarrow  b_i = m-2 \\  
c_i = m - 1 &\longrightarrow  b_i = m -1 \\ 
c_i = m &\longrightarrow b_i = m - 1 \\
c_i = m + 1 &\longrightarrow b_i = m .
\end{align*}  
Then $g_m$ is a bijection.
\end{lemma}

\begin{proof}

It is clear from the definition that $g_m$ is a function.  So we first show that $g_m$ is one-to-one. Suppose $x = [x_r, x_{r-1}, \ldots x_2, x_1, x_0]_{m+1}$ and $y = [y_r, y_{r-1}, \ldots y_2, y_1, y_0]_{m+1}$ are two hyper $(m+1)$-ary partitions of $k$ such that $x \neq y$. Then there must be at least one digit that doesn't match. Let $J = \{ j_1, j_2, \dots, j_n \}$ be the set of indices such that $x_j \neq y_j$. Then we have two cases. 

First suppose without loss of generality that there is an index $j$ such that $x_j \notin \{ m-1, m \}$. Then the $j^{th}$ digit of $g_m(x)$ will be different than the $j^{th}$ digit of $g_m(y)$. Thus $g_m(x) \neq g_m(y)$. 

Now suppose that $x_j \in \{ m-1,m \}$ and $y_j \in \{ m-1,m \}$ for all $j \in J$. Let $J_1 = \{ j \in J: x_j = m-1 \}$ and $J_2 = \{ j \in J: x_j = m \}$. Note that $y_j = m$ for all $j \in J_1$ and $y_j = m-1$ for all $j \in J_2$. Also observe that
\begin{eqnarray*}
	x &=& \sum_{j \notin J} x_j m^j + \sum_{j \in J_1} (m-1) m^j + \sum_{j \in J_2} m \cdot m^j \\
	y &=& \sum_{j \notin J} y_j m^j + \sum_{j \in J_1} m \cdot m^j + \sum_{j \in J_2} (m-1) m^j. 
\end{eqnarray*}
Since $x_j = y_j$ for all $j\notin J$, 
\begin{eqnarray*}
	x - y &=& \sum_{j \in J_1} (m-1-m) m^j + \sum_{j \in J_2} (m-m+1) m^j \\
				&=& \sum_{j \in J_2} m^j - \sum_{j \in J_1} m^j.
\end{eqnarray*}
Observe that $x-y = 0$ since $x$ and $y$ are two different hyper $(m+1)$-ary partitions of the same number $k$, implying 
\[
  \sum_{j \in J_2} m^j - \sum_{j \in J_1} m^j = 0.
\]
However, since $J_1$ and $J_2$ are disjoint, this is impossible. Thus it must be the case that when $x \neq y$, one of $x_j$ or $y_j$ must be outside of $\{ m-1, m \}$ so that $g_m(x) \neq g_m(y)$ as seen above. Thus $g_m$ is one-to-one.

To show that $g_m$ is onto, consider $b = [b_r, b_{r-1}, \cdots, b_2, b_1, b_0]_m \in H_m(\ell)$. We then define ${c=[c_r, c_{r-1}, \cdots, c_2, c_1, c_0]_{m+1}}$ in the following way. If $b_i \in \{ 0,1,2,\dots,m-3,m-2 \}$, then set $c_i = b_i$ and if $b_i=m$, set $c_i=m+1$. Now suppose $b_i = m-1$. Let $v$ be the minimal index with $v<i$ such that $b_v \neq m-1$. If $v$ does not exist, then set $c_i = m-1$. If $v$ does exist with $b_v = m$, then set $c_i=m$. If $v$ exists with $b_v \in \{0,1,2,\dots,m-2\}$, then set $c_i = m-1$. Notice that we may verify that $c\in H_{m+1}(k)$ by converting $c$ into the base $m+1$ representation of $k$. Therefore $b$ is the image of $c$ under $g_m$ and thus $g_m$ is onto.
\end{proof}

This bijection implies that the number of $m$-ary partitions of any integer $\ell$ is the same as the number of $(m+1)$-ary partitions of $F_{m,m+1}(\ell)$.


\section{Hyper $m_1$-ary partitions and hyper $m_2$-ary partitions}

In this section, we use the result of Lemma \ref{mtom+1_bijection} to define a more general bijection between $H_{m_1}(n)$ and $H_{m_2}(F_{m_1,m_2}(n))$ for $m_2 > m_1 + 1$.  To do this, we need the following lemma about hyper $m_2$-ary partitions of an integer $n$.

In the following proof, observe that multiplying a partition $[x_r, x_{r-1}, \cdots, x_2, x_1, x_0]_m$ by $m$ corresponds to shifting the coefficients to the left one place and adding an additional 0 as the last coefficient. 
 
\begin{lemma}\label{which_digitsM}
Let $m_2 > m_1 + 1$.  If the base $m_2$ representation of an integer $n$ contains only digits from the set $\{ 0,1,2,\ldots, m_1 -1 \}$, then there are no hyper $m_2$-ary partitions of $n$ which use any of the coefficients $m_1, m_1 + 1, \ldots, m_2 - 2$.
\end{lemma}
\begin{proof}

We will prove this by induction on $n$. Assume that for all $q<n$, when the base $m_2$ representation of $q$ contains only digits from the set $\{ 0,1,2,\ldots, m_1 -1 \}$, then there are no hyper $m_2$-ary partitions of $n$ which use any of the coefficients $m_1, m_1 + 1, \ldots, m_2 - 2$.

First, consider when $n=m_2q$ and suppose that in base $m_2$ the digits of $n$ come from the set $\{ 0,1,2,\ldots, m_1 -1 \}$.  This means the digits in the base $m_2$ representation of $q$ also come only from this set.  Now, apply the recurrence (\ref{recur1}) to write $h_{m_2}(m_2q) = h_{m_2}(q) + h_{m_2}(q-1)$.  This implies that every hyper $m_2$-ary partition of $n$ is obtained from either a hyper $m_2$-ary partition of $q$ or a hyper $m_2$-ary partition of $q-1$. 

Observe that a hyper $m_2$-ary partition of $n$ obtained from a hyper $m_2$-ary partition of $q$ is found by multiplying the latter partition by $m_2$, thereby shifting the coefficients of $q$ and appending a 0 at the end. This results in hyper $m_2$-ary partitions of $n$ whose coefficients are the same as the coefficients of hyper $m_2$-ary partitions of $q$, along with an additional 0. Similarly, a hyper $m_2$-ary partition of $n$ that is obtained from a hyper $m_2$-ary partition of $q-1$ is found by shifting the digits of the latter partition and appending an $m_2$ to the end. This means we may write
\begin{align*}
H_{m_2}(n) = &\left\{ [x_r, x_{r-1}, \cdots, x_2, x_1, x_0, 0]_{m_2} : [x_r, x_{r-1}, \cdots, x_2, x_1, x_0]_{m_2} \in H_{m_2}(q) \right\} \\
&\cup \left\{ [x_r, x_{r-1}, \cdots, x_2, x_1, x_0, m_2]_{m_2} : [x_r, x_{r-1}, \cdots, x_2, x_1, x_0]_{m_2} \in H_{m_2}(q-1) \right\}.
\end{align*}

Since $q-1$ and $q$ are less than $n$, by the induction hypothesis we know the coefficients of all hyper $m_2$-ary partitions of $q-1$ and $q$ are from the set $\{ 0,1,2,\ldots, m_1 -1, m_2 -1, m_2 \}$.  Thus, the coefficients of any hyper $m_2$-ary partition of $n$ are also from this set.  

Now assume that $n=m_2q+s$, where $1 \le s \leq m_2-1$. Observe that since the base $m_2$ representation of $n$ contains only digits from the set $\{0,1,2,\dots,m_1-1\}$, then we must have $1 \le s \leq m_1-1$. Furthermore, when $n=m_2q+s$, apply the recurrence (\ref{recur2}) to conclude that a hyper $m_2$-ary partition of $n$ is obtained from a hyper $m_2$-ary partition of $q$ by multiplying the latter partition by $m_2$ and appending $s$ to the end, where $1 \le s \leq m_1-1$.  So,
$$H_{m_2}(n) = \left\{ [x_r, x_{r-1}, \cdots, x_2, x_1, x_0, s]_{m_2} : [x_r, x_{r-1}, \cdots, x_2, x_1, x_0]_{m_2} \in H_{m_2}(q) \right\}.$$
Since $q<n$, the coefficients of all hyper $m_2$-ary partitions of $q$ are in the set $\{ 0,1,2,\ldots, m_1 -1, m_2 -1, m_2 \}$.  Since $s$ is an element of this set, we conclude that the coefficients of hyper $m_2$-ary partitions of $n$ come from the same set. 

Therefore, in all cases, the hyper $m_2$-ary partitions of $n$ never contain any of the coefficients $m_1, m_1 + 1, \ldots, m_2 - 2$.
\end{proof}

Now we are ready to prove there is a bijection between hyper $m_1$-ary partitions of an integer $\ell$ and hyper $m_2$-ary partitions of $k = F_{m_1,m_2}(\ell)$.  

\begin{lemma}\label{general_theorem}
Let $\ell$ be a positive integer and set $k = F_{m_1,m_2}(\ell)$.  Define $\phi: H_{m_2}(k) \rightarrow H_{m_1}(\ell)$ by mapping
$$[c_r, c_{r-1}, \ldots, c_2 ,c_1 ,c_0]_{m_2} \mapsto [b_r ,b_{r-1}, \cdots, b_2, b_1, b_0]_{m_1}$$
according to the following rules:
\begin{align*}
c_i  = 0 &\longrightarrow  b_i = 0  \\ 
c_i = 1  &\longrightarrow b_i = 1  \\ 
& \, \, \, \,\,  \vdots \\
c_i = m_1 -1 &\longrightarrow  b_i = m_1 -1 \\  
c_i = m_2 -1 &\longrightarrow  b_i = m_1 -1 \\ 
c_i = m_2 &\longrightarrow b_i = m_1 .
\end{align*}  
Then, $\phi$ is a bijection.
\end{lemma}
\begin{proof}
If $m_2 = m_1 + 1$, then the result follows immediately from Lemma \ref{mtom+1_bijection}.  So, we assume that $m_2 > m_1 + 1$.  From the definition of $k$, we know the base $m_2$ representation of $k$ includes only digits less than or equal to $m_1 - 1$.  So, we apply Lemma \ref{which_digitsM} to conclude that none of the hyper $m_2$-ary partitions in $H_{m_2}(k)$ have any coefficients between $m_1$ and $m_2 - 2$, inclusive.  Thus, $\phi$ need only specify how to map coefficients from the set $\{0,1, \ldots, m_1 - 1, m_2 - 1, m_2 \}$. 

Now, using the bijection $g_m$ given in Lemma \ref{mtom+1_bijection}, define a new function $G:H_{m_2}(k) \rightarrow H_{m_1}(\ell)$ as follows:
$$ G = g_{m_1} \circ g_{m_1 + 1} \circ g_{m_1 + 2} \circ \cdots \circ g_{m_2 - 2} \circ g_{m_2 - 1}  \ .$$  
It is clear from Lemma \ref{mtom+1_bijection} that when we apply $G$ to any $m_2$-ary partition coefficient which is less than or equal to $m_1 - 1$, the coefficient maps to itself.  When we apply $G$ to a partition coefficient of $m_2 - 1$, we see that 
$$ m_2 - 1 \xmapsto{g_{m_2 -1}} \, m_2 - 2 \xmapsto{g_{m_2 -2}} \,  m_2 - 3 \xmapsto{g_{m_2 -3}} \, \cdots \, \xmapsto{g_{m_1}} m_1 - 1 \ .$$
Finally, when we apply $G$ to a partition coefficient of $m_2$, we see that 
$$ m_2  \xmapsto{g_{m_2 -1}} \, m_2 - 1 \xmapsto{g_{m_2 -2}} \, m_2 - 2 \xmapsto{g_{m_2 -3}} \, \cdots \, \xmapsto{g_{m_1}} 
m_1  \ .$$
Thus, $G = \phi$. 

We have $\phi$ equal to a finite composition of bijective functions.  Therefore, $\phi$ is a bijection.
\end{proof}
Lemma \ref{general_theorem} leads to the following identity between values of $h_{m_1}$ and $h_{m_2}$. 

\begin{theorem}
	Let $2 \leq m_1 < m_2$.  For positive integer $\ell$, set $k = F_{m_1,m_2}(\ell)$.	Then
	\[
		h_{m_2}(k) = h_{m_1}(\ell).
	\]
\end{theorem}

\begin{proof}
The values $\ell$ and $k$ given here match Lemma \ref{general_theorem} and we know that $h_{m_1}(\ell) = | H_{m_1}(\ell) |$ and $h_{m_2}(k) = | H_{m_2}(k) |$.  Lemma \ref{general_theorem} gives a bijection between these finite sets.  Therefore, we conclude that the sets must have the same cardinality.
\end{proof}
As an immediate corollary, we now state a final result regarding the relationships between hyper $m$-ary partition sequences for different values of $m$. 
\begin{corollary}
Let $2 \leq m_1 \leq m_2$.  Then $h_{m_1}$ is a subsequence of $h_{m_2}$.
\end{corollary}

These theorems extend the results in \cite{ours}, ultimately showing that the subsequence identity holds for any hyper $m_1$-ary and hyper $m_2$-ary partition sequences.

\section{Acknowledgments}

The authors wish to thank the referee for insightful comments and suggestions.  This feedback helped us clarify the notation and improved the overall quality of the paper.


\begin{thebibliography}{9}

\bibitem{recounting} N. Calkin and H. Wilf, Recounting the rationals,
{\it Amer. Math. Monthly} {\bf 107} (2000), 360--363.

\bibitem{m-ary} K. Courtright and J. Sellers, Arithmetic properties for
hyper $m$-ary partition functions, {\it Integers} {\bf 4} (2004), A6.

\bibitem{sohn} J. Eom, G. Jeong, and J. Sohn, Three different ways to
obtain the values of hyper $m$-ary partition functions, {\it Bull.
Korean Math. Soc.} {\bf 53} (2016), 1857--1868.

\bibitem{ours} T. B. Flowers and S. R. Lockard, Identifying an $m$-ary
partition identity through an $m$-ary tree, {\it Integers} {\bf 16}
(2016), A10.

\bibitem{northshield} S. Northshield, Stern's diatomic sequence
$0,1,1,2,1,3,2,3,1,4,\ldots$, {\it Amer. Math. Monthly} {\bf 117}
(2010), 581--598.

\bibitem{reznick} B. Reznick, Some binary partition functions, in B.
Berndt, ed., {\it Analytic Number Theory: Proceedings of a Conference
in Honor of Paul T. Bateman}, Birkh{\"a}user, 1990, pp.~451--477.

\bibitem{OEIS} N. J. A. Sloane, The On-Line Encyclopedia of Integer
Sequences. Published electronically at \url{http://oeis.org}, 2017.

\bibitem{stern} M. A. Stern, {\"U}ber eine zahlentheoretische Funktion,
{\it J. Reine Angew. Math.} {\bf 55} (1858), 193--220.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary 05A17.  \\
\noindent \emph{Keywords: } integer partition, hyper $m$-ary partition.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A002487},
\seqnum{A054390},
\seqnum{A277872}, and
\seqnum{A277873}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received  June 30 2016;
revised versions received  February 9 2017; June 13 2017; June 23 2017.
Published in {\it Journal of Integer Sequences}, July 1 2017.

\bigskip
\hrule
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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