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\begin{document}

\begin{center}
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\begin{center}
\vskip 1cm{\LARGE\bf A Partition Formula for Fibonacci Numbers
}
\vskip 1cm
\large
Philipp Fahr and Claus Michael Ringel \\ 
Fakult\"at f\"ur Mathematik\\ 
Universit\"at Bielefeld\\ 
P. O. Box 100 131\\ 
D-33501 Bielefeld \\
Germany\\
\href{mailto:philfahr@math.uni-bielefeld.de}{\tt philfahr@math.uni-bielefeld.de} \\
\href{mailto:ringel@math.uni-bielefeld.de}{\tt ringel@math.uni-bielefeld.de} \\
\end{center}

\vskip .2 in

\begin{abstract}
We present a partition formula for the even index Fibonacci numbers.
The formula is motivated by the appearance of these Fibonacci numbers in the
representation theory of the socalled $3$-\-Kro\-necker quiver, i.e., the oriented graph with two vertices and three arrows in the same direction.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}



\section{Introduction}

Let $f_0,f_1,\dots$ be the Fibonacci numbers, with $f_0 = 0, f_1 = 1$
and $f_{i+1} = f_i + f_{i-1}$ for $i \ge 1.$ The aim of this note is to
present a partition formula for the even index Fibonacci numbers $f_{2n}$.
Our interest in the even index Fibonacci numbers comes from the fact that the pairs $[f_{4t+2},f_{4t}]$ provide the dimension vectors of the preprojective 
representations of 3-Kronecker quiver, and the partition formula relates such a dimension
vector to the dimension vector of a corresponding graded representation. 
The authors are grateful to M. Baake and A. Zelevinsky for useful comments
which have been incorporated into the paper.

We start with the 3-regular tree $(T,E)$ with vertex set $T$ and edge set $E$
(3-regularity means that every vertex of $T$ has precisely 3 neighbours). Fix a vertex
$x_0$ as base point. Let $T_r$ be the sets of vertices of $T$ which
have distance $r$ to $x_0$, thus $T_0 = \{x_0\}$, $T_1$ are the neighbours
of $x_0$, and so on. The vertices in $T_r$ will be called even or odd, depending on $r$ being even or odd. Note that $|T_0| = 1$ and 
\begin{equation}
 |T_r| = 3\cdot 2^{r-1} \qquad \text{for } r \ge 1. \tag{1}
\end{equation}
(The proof for equation $(1)$ is by induction: $T_1$ consists of the three neighbours of
$x_0$. For $r \ge 1$, any element in $T_r$ has precisely one neighbour
in $T_{r-1}$, thus two neighbours in $T_{r+1},$  therefore $|T_{r+1}| =
2|T_r|.$)

Given a set $S$, let $\mathbb{Z}[S]$ be the set of 
functions $S \to \mathbb{Z}$ with finite support; this is the free abelian group
on $S$ (or better, on the set of functions with support 
being an element of $S$ such that the only non-zero value is $1$).

We are interested in certain elements $a_t \in \mathbb{Z}[T]$. 
Any vertex $y\in T$ yields a reflection $\sigma_y$ on $\mathbb{Z}[T]$, defined
for $b\in \mathbb{Z}[T]$ by 
$$  (\sigma_y b)(x) =   \left\{ \begin{array}{lll}
b(x) &amp; &amp; x \neq y\\
&amp; \textrm{provided}\\
-b(y) + \sum\noindent_{z\in N(y)} b(z) &amp; &amp; x = y,
\end{array} \right.$$
where $N(y)$ denotes the set of neighbours of $y$ in $(T,E)$.

Denote by $\Phi_0$ the product of the reflections $\sigma_y$ with $y$ even;
this product is independent of the order, since these reflections commute:
even vertices never are neighbours of each other.
Similarly, we denote by $\Phi_1$ the product of 
the reflections $\sigma_y$ with
$y$ odd. 

The elements of $\mathbb{Z}[T]$ we are interested in are labelled $a_t$ with
$t\in \mathbb{N}_0.$ 
We will start with the characteristic function $a_0$ of $T_0$
(thus, $a_0(x) = 0$ unless $x = x_0$ and $a_0(x_0) = 1$) and look at 
the functions
$$  a_t = (\Phi_0\Phi_1)^t a_0, \quad \text{with } t \ge 0. $$
Clearly, $a_t$ is constant on $T_r$, for any $r\ge 0$, so we may
define $a_t: \mathbb{N}_0 \to \mathbb{Z}$ by
$$  a_t[r] = a_t(x) \quad \text{for } r\in \mathbb{N}_0 \text{ and } x \in T_r. $$

\section{The Partition Formula}

\begin{center} 
%\fbox
{ \begin{eqnarray*}
 f_{4t}   &amp;=&amp; \sum_{r \text{ odd }} |T_r|\cdot a_t[r] 
  = \; \ \qquad \ \ \ 3\sum_{m\ge 0} 2^{2m}\cdot a_t[2m\!+\!1], \\
 f_{4t+2} &amp;=&amp; 
\sum_{r \text{ even}} |T_r|\cdot a_t[r] 
  = \ a_t[0] + 3\sum_{m\ge 1} 2^{2m-1}\cdot a_t[2m]. 
\end{eqnarray*}
}
\end{center}
The sums exhibited are finite sums, since $a_t[r] = 0$ for $r &gt; 2t.$ 
Note that we have used the equality (1).


For example, for $t=3$, we obtain the following two equalities:
\begin{eqnarray*}
 144 = f_{12} &amp;=&amp;  3\cdot 12 + 12\cdot 5 + 48 \cdot 1 \cr
 377 = f_{14} &amp;=&amp; 29 + 6\cdot 18 + 24\cdot 6 + 96 \cdot 1.
\end{eqnarray*}

Here is the table for $t\le 6$.
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\section{Proof of the partition formula}

We consider the multigraph $(\overline T,\overline E)$ with
two vertices $0,1$ and three edges between $0$ and $1$. 

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Any element $c \in \mathbb{Z}[\overline T]$ is determined by the 
values $c(0)$ and $c(1)$, thus we will write $c = [c(0),c(1)]$.\\

We define 
\begin{eqnarray*}
\overline a_t(0) &amp;=&amp; \sum_{r \text{ even}} |T_r|\cdot a_t[r] 
  = \ a_t[0] + 3\sum_{m\ge 1} 2^{2m-1}a_t[2m],  \\
 \overline a_t(1) &amp;=&amp; \sum_{r \text{ odd }} |T_r|\cdot a_t[r] 
  = \; \ \qquad \ \ \ 3\sum_{m\ge 0} 2^{2m}a_t[2m\!+\!1],
\end{eqnarray*}
and have to show: $$[\overline a_t(0),\overline a_t(1)] = [f_{4t+2},f_{4t}].$$
The claim is obviously true for $t= 0$, since $$[\overline a_0(0),\overline a_0(1)] = [1,0] = [f_{2},f_{0}].$$
The general assertion will follow from the following  recursion formulae
$$[\overline a_t(0),\overline a_t(1)] \MATRIX{8}{3}{-3}{-1} = [\overline a_{t+1}(0),\overline a_{t+1}(1)],$$
$$[f_{m+2},f_{m}] \MATRIX{8}{3}{-3}{-1}= [f_{m+6},f_{m+4}].$$
For the Fibonacci numbers, this is well-known:
\begin{eqnarray*} 
 [f_{m+6},f_{m+4}] &amp;= &amp;[f_{m+5},f_{m+4}] \MATRIX{1}{0}{1}{1} \\
&amp;=&amp;  [f_{m+1},f_m] \MATRIX{1}{1}{1}{0}^4 \MATRIX{1}{0}{1}{1}\\
&amp; =&amp; [f_{m+2},f_m]   \MATRIX{1}{0}{1}{1}^{-1} \MATRIX{1}{1}{1}{0}^4 \MATRIX{1}{0}{1}{1}
\end{eqnarray*}
and the product of the matrices in the last line
is just $\MATRIX{8}{3}{-3}{-1}$, which is the Coxeter transformation of the 3-Kronecker quiver.

It remains to deal with $\overline a_t.$ 
We may consider
$(T,E)$ as the universal covering of $(\overline T,\overline E)$, 
say with a covering map 
$$
 \pi:(T,E) \to (\overline T,\overline E),
$$ 
where $\pi(x) = 0$ if $x$
is an even vertex of $(T,E)$, and $\pi(x) = 1$ if $x$ is an odd vertex.
Such a covering map will provide a bijection between 
the three edges in $E$ which
contain a given vertex $x$ with the edges in $\overline E.$ 

Given $b\in \mathbb{Z}[T]$, define $\overline b\in \mathbb{Z}[\overline T]$
by
$$
 \overline b(i) = \sum\nolimits_{x\in \pi^{-1}(i)} b(x) \qquad \text{for }
i = 0,1.
$$
Observe that in this way we obtain from $a_t \in \mathbb{Z}[T]$ precisely
$\overline a_t \in \mathbb{Z}[\overline T]$ as defined above.

On $\mathbb{Z}[\overline T]$, we also consider reflections, but we have to take
into account the multiplicity of edges: There are the two reflections
$\sigma_0, \sigma_1$, with
\begin{eqnarray*}
 (\sigma_0c)(0) =&amp; -c(0)+3c(1),\qquad &amp; (\sigma_0c)(1) =c(1), \\
 (\sigma_1c)(0) =&amp; c(0),    \qquad &amp;  (\sigma_1c)(1) = -c(1)+3c(0),
\end{eqnarray*}
for $c\in \mathbb{Z}[\overline T]$.
Note that this implies that
$$ [c(0),c(1)] \bmatrix 8 &amp; 3 \cr
         -3 &amp; -1 \endbmatrix 
 = [(\sigma_0\sigma_1c)(0),(\sigma_0\sigma_1c)(1)].$$
We finish the proof by observing that $\overline a_t = (\sigma_0\sigma_1)^t\overline a_0.$
This follows directly from the following consideration: Let $b\in 
\mathbb{Z}[T]$, then
$$ \overline{\Phi_0b} = \sigma_0\overline b,\quad 
 \overline{\Phi_1b} = \sigma_1\overline b,$$
as it is easily verified.

\section{A second approach}

A. Zelevinsky has pointed out that it may be worthwhile to stress  the following
separation: Let us denote by $b_t[r]$ and $c_t[r]$ the sequences
$$
 b_t[r] = a_t[2r] \quad \text{and} \quad c_t[r] = a_{t+1}[2r+1].
$$
Then we obtain two ``Pascal-like'' triangles
$$
  \bmatrix b_0[0] &amp; b[1]_0 &amp; b_0[2] &amp; \cdots  \cr
              b_1[0] &amp; b_1[1] &amp; b_1[2] &amp; \cdots  \cr
              b_2[0] &amp; b_2[1] &amp; b_2[2] &amp; \cdots  \cr
              b_3[0] &amp; b_3[1] &amp; b_3[2] &amp; \cdots  \cr
              b_4[0] &amp; b_4[1] &amp; b_4[2] &amp; \cdots  \cr
              &amp; \cdots 
\endbmatrix 
= \bmatrix 1 \cr
           2 &amp; 1 \cr
           7 &amp; 4 &amp; 1 \cr
           29 &amp; 18 &amp; 6 &amp; 1 \cr
           130 &amp; 85 &amp; 33 &amp; 8 &amp; 1 &amp;  \cr
           &amp; \cdots
\endbmatrix
$$
and 
$$
 \bmatrix c_0[0] &amp; c_0[1] &amp; c_0[2] &amp; \cdots  \cr
              c_1[0] &amp; c_1[1] &amp; c_1[2] &amp; \cdots  \cr
              c_2[0] &amp; c_2[1] &amp; c_2[2] &amp; \cdots  \cr
              c_3[0] &amp; c_3[1] &amp; c_3[2] &amp; \cdots  \cr
              c_4[0] &amp; c_4[1] &amp; c_4[2] &amp; \cdots  \cr
              &amp; \cdots 
\endbmatrix 
= \bmatrix 1 \cr
           3 &amp; 1 \cr
           12 &amp; 5 &amp; 1 \cr
           53 &amp; 25 &amp; 7 &amp; 1 \cr
           247 &amp; 126 &amp; 42 &amp; 9 &amp; 1 &amp;  \cr
           &amp; \cdots
\endbmatrix
$$
The recursive definition is as follows: One starts with 
$$
 b_0[r] = c_0[r] = \delta_{r,0}
$$
(using the Kronecker delta) and one continues for $t \ge 0$ with
$$
\begin{matrix}
 b_{t+1}[r] &amp;= &amp; \!\!\!\!\!\!\!\! c_t[r-1] + 2 c_t[r] - b_t[r], \cr
 c_{t+1}[r] &amp;= &amp; b_{t+1}[r]+ 2 b_{t+1}[r+1] - c_t[r], \cr
\end{matrix}
$$
where $c_t[-1] = c_t[0]$.

Using this notation, the partition formula can be written in the following way:

 \begin{center} \fbox{ 
$  f_{4t+2} = b_t[0]+ 3\sum_{r=1}^t 2^{2r-1}\cdot b_t[r],\qquad 
 f_{4(t+1)} = 3\sum_{r=0}^t 2^{2r}\cdot c_t[r].
$ } \end{center}



\section{Remarks} 
\indent \indent 1. The partition formula presented here seems to be
very natural. We were surprised to see that the integer sequence $a_t[0]$
had not been listed in Encyclopedia of Integer Sequences \cite{Sloane}.
It is now recorded as sequence \seqnum{A132262}. There seems to be numerical evidence that the sequence $a_t[1]$ 
is just the sequence \seqnum{A110122} of the Encyclopedia of Integer Sequences (it counts the Delannoy paths of length $t$ with no EE's crossing the line $y=x$), but we do not see why
these sequences are the same. \\

2. The 3-regular tree which is used to depict the partition formula
plays a prominent role in many parts of mathematics. It is the 3-Cayley
tree wihout leafs, and a Bruhat-Tits tree. As M. Baake has pointed out to us, 
it is called the Bethe lattice
\cite{Ba} with coordination number 3 in the theory of exactly solvable models.\\

3. A similar method as presented here leads to   
a partition formula for the odd index Fibonacci numbers, see \cite{F}.\\

4. Our considerations can be generalized to the $n$-regular tree and the multigraph with two vertices and $n$ edges, i.e., to the $n$-Kronecker quiver.

\section{Motivation and background}

Let us recall some basic concepts from the representation theory of quivers
(see for example
\cite{ARS}); note that quivers are just directed graphs were mutiple arrows and even loops are allowed.
Given any quiver $Q$, one considers its representations with coefficients in a fixed field $k$:
they are obtained by attaching to each vertex $a$ of $Q$ a (usually finite-dimensional)
$k$-vector space $V_a$, and to each arrow $\alpha: a \to b$ a $k$-linear map $V_\alpha :
V_a \to V_b$. The function which assigns to the vertex $a$ the $k$-dimension of $V_a$ is
called the dimension vector $\mathbf{dim}\,V$  of $V$; in case $V_a = 0$ for almost all vertices
$a$, we obtain an element of $\mathbb Z[Q_0]$, where $Q_0$ is the vertex set of $Q$, 
and $\mathbb Z[Q_0]$ itself may be interpreted
as the Grothendieck group of the finite dimensional nilpotent 
representations of $Q$ with respect to all exact sequences.
Observe that the representations 
of a quiver form an abelian category and one is interested in the
indecomposable objects of this category. In case we deal with a finite quiver without loops,
Kac \cite{K2} has shown that the dimension vectors of the indecomposable objects 
are precisely the positive roots $\mathbf v$ of the Kac-Moody Lie-algebra defined by the
underlying graph of the quiver.
In case $\mathbf v$ is a real root, then there is precisely one isomorphism class, otherwise there are at least two (actually infinitely many, in case $k$ itself is infinite). 

The quivers which are of interest in this note are, on the one hand, the 3-Kronecker quiver 
$\overline Q$ (with two vertices, say labelled $0$ and $1$, and three arrows $1 \to 0$) and, on the other hand, the bipartite quiver $Q$ whose underlying graph (obtained by deleting the orientation of the arrows) is the 3-regular tree.
The Kac-Moody Lie-algebra \cite{K1} corresponding to the 3-Kronecker quiver is given by the generalized Cartan matrix $\MATRIX{2}{-3}{-3}{2}$, its positive real roots are of the form 
$$
 [f_{4t+2},f_{4t}],\ \text{and} \ [f_{4t},f_{4t+2}] \ \text{with} \ t\ge 0.
$$ 
Note that the real roots of a Kac-Moody Lie-algebra are always obtained from the simple
roots by reflections; in the given case, one uses powers of the Coxeter transformation 
$\MATRIX{8}{3}{-3}{-1}$ in order to create the positive roots of the form $[f_{4t+2},f_{4t}]$
starting with the vectors $[1,0]$ and $[3,1]$. 
Alternatively, one may stress that the square
$\MATRIX{2}{1}{1}{1}$ of the matrix
which exhibits the Fibonacci sequence, is
conjugate to the matrix $\MATRIX{3}{1}{-1}{0}$
which describes a basic reflection for the 3-Kronecker
quiver.

There is a 
functorial realisation of the reflections as well as of the Coxeter
transformation by Bernstein-Gelfand-Ponomarev \cite{BGP}.
The indecomposable representations of the 3-Kronecker quiver with dimension vector
$[1,0]$ and $[3,1]$ are just the projective ones, those with dimension vectors of the
form $[f_{4t+2},f_{4t}]$ are said to be preprojective. Instead of dealing with the Coxeter functors of Bernstein-Gelfand-Ponoarev, one also may use the Auslander-Reiten translation functor
(see \cite{ARS}) in order to construct the preprojective representations starting from the
projective ones. The preprojective representations have been exhibited in matrix form
by one of the authors \cite{RingelTree}. 

It remains to mention the covering theory of Gabriel and his students \cite{G}. The quiver
$Q$ may be considered as the universal covering of the quiver $\overline Q$. Any representation $V$ of $Q$ gives rise to a representation $\overline V$ of $\overline Q$ (by attaching to the vertex 0 of
$\overline Q$ the direct sum of the vector spaces attached to the various sinks of $Q$, and
attaching to  1  the direct sum of the vector spaces attached to the sources). The representations of 
$\overline Q$ obtained in this way are those which are gradable by the free (non-abelian)
group in 3 free generators. The covering functor $V \mapsto \overline V$ preserves
indecomposability and satisfies
$$
 \mathbf{dim}\,\overline V = \overline{\mathbf{dim}\,V}. \qquad (*)
$$ 
Any preprojective representation of $\overline Q$ is of the form $\overline V$, where $V$ is
an indecomposable representation of $Q$ which is obtained from a simple representation
of $Q$ by applying suitable reflection functors, and our partition formula is just the
assertion $(*)$. 


\section{Illustrations}
The illustrations below show the functions $a_t$ for $t=0,1,2$ as well
as the corresponding functions $\Phi_1a_t$.
Here, $(T,E)$ has been endowed with an orientation such that the edges point to the even
vertices. In this way, the even vertices are sinks, the odd vertices are
sources. The dotted circles indicate the various sets $T_r$, the
center is just $T_0 = \{x_0\}.$




$$
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%=============================================2.Arm
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%===============================================
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%=======================================================

%=============================================2.Arm
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%===============================================
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%=======================================================


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2.Bild
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%===============================================
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%=============================================2.Arm
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%===============================================
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%=======================================================

%=============================================2.Arm
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%===============================================
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%=======================================================


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3.Bild
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%===============================================
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%=============================================2.Arm
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%===============================================
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%=======================================================

%=============================================2.Arm
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%===============================================
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%=======================================================


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%4.Bild
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%=============================================2.Arm
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%===============================================
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$$


\begin{thebibliography}{99}
\bibitem{ARS}
M. Auslander, I. Reiten and S. Smal\o, 
{\em Representation theory of Artin algebras.}
Cambridge Studies in Advanced Mathematics, 1994.

\bibitem{Ba}
R. J. Baxter, {\em Exactly Solved Models in Statistical Mechanics.} 
Academic Press, 1982.

\bibitem{BGP}
I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev,
Coxeter functors and Gabriel's theorem.
{\em Uspechi Mat. Nauk.} \textbf{28} (1973), 19--33; {\em Russian Math. Surveys} \textbf{29} (1973), 17--32.


\bibitem{G}
P. Gabriel, {\em The universal cover of a representation-finite algebra.} 
In {\it Representations of Algebras, Puebla (1980)}, 
Lecture Notes in Mathematics, Vol. 903, Springer, 1981, pp.\ 68--105.

\bibitem{F}
Ph. Fahr, Infinite Gabriel-Roiter measures for the 3-Kronecker quiver.
{\em Dissertation Bielefeld}, in preparation.

\bibitem{K1}
V. G. Kac, {\em Infinite-Dimensional Lie Algebras.}
3rd edition, Cambridge University
Press, 1990.

\bibitem{K2}
V. G. Kac, {\em Root systems, representations of quivers and invariant theory.} {\it Invariant theory, Montecatini (1982)},
Lecture Notes in Mathematics, Vol.\ 996, Springer, 1983, pp.\ 74--108.

\bibitem{Ko}
Th. Koshy, {\em 
Fibonacci and Lucas Numbers with Applications.}
Wiley Series in Pure and Applied Mathematics, 2001.

\bibitem{RingelTree}
C. M. Ringel, Exceptional modules are tree modules.
{\em Linear Algebra and its applications} (1998), 471--493.

\bibitem{Sloane}
N. J. A. Sloane, 
{\em The On-Line Encyclopedia of Integer Sequences}, 
available electronically at
\href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/$\sim$njas/sequences/}.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B39, Secondary 16G20.

\noindent \emph{Keywords: } 
Fibonacci numbers, universal cover, 3-regular tree, 3-Kronecker quiver.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences \seqnum{A110122} and \seqnum{A132262}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received August 13 2007;
revised version received January 22 2008.  
Published in {\it Journal of Integer Sequences}, February 9 2008.

\bigskip
\hrule
\bigskip

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


\end{document}

                                                                                

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