A Partition Formula for Fibonacci Numbers

Philipp Fahr and Claus Michael Ringel Fakultät für Mathematik Universität Bielefeld P. O. Box 100 131 D-33501 Bielefeld Germany

                philfahr@math.uni-bielefeld.de
                 ringel@math.uni-bielefeld.de

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-Kronecker quiver, i.e., the oriented graph with two vertices and three arrows in the same direction.

1 Introduction

Let f_{0}, f_{1}, … be the Fibonacci numbers, with f_{0} = 0, f_{1} = 1 and f_{i+1} = f_{i} + f_{i−1} for i ≥ 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

|T_{r}| = 3 · 2^{r−1} for r ≥ 1.  (1)

(The proof for equation (1) is by induction: T_{1} consists of the three neighbours of x_{0}. For r ≥ 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 ℤ[S] be the set of functions S → 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} ∈ ℤ[T]. Any vertex y ∈ T yields a reflection σ_{y} on ℤ[T], defined for b ∈ ℤ[T] by

(σ_{y}b)(x) = \{ \begin{matrix}b(x) & & x ≠ y \\ & provided \\ −b(y) + ∑_{z∈N(y)} b(z) & & x = y, \end{matrix}

where N(y) denotes the set of neighbours of y in (T, E).

Denote by Φ_{0} the product of the reflections σ_{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 Φ_{1} the product of the reflections σ_{y} with y odd.

The elements of ℤ[T] we are interested in are labelled a_{t} with t ∈ 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} = (Φ_{0}Φ_{1})^{t}a_{0}, with t ≥ 0.

Clearly, a_{t} is constant on T_{r}, for any r ≥ 0, so we may define a_{t} : ℕ_{0} → ℤ by

a_{t}[r] = a_{t}(x) for r ∈ ℕ_{0} and x ∈ T_{r}.

2 The Partition Formula

f_{4t} = ∑_{r odd} |T_{r}| · a_{t}[r] =  3 ∑_{m≥0} 2^{2m} · a_{t}[2m+1],
f_{4t+2} = ∑_{r even} |T_{r}| · a_{t}[r] = a_{t}[0] + 3 ∑_{m≥1} 2^{2m−1} · a_{t}[2m].

The sums exhibited are finite sums, since a_{t}[r] = 0 for r > 2t. Note that we have used the equality (1).

For example, for t = 3, we obtain the following two equalities:

144 = f_{12} = 3 · 12 + 12 · 5 + 48 · 1
377 = f_{14} = 29 + 6 · 18 + 24 · 6 + 96 · 1.

Here is the table for t ≤ 6.

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987 2584 6765 17711 46368 121393

3 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 ∈ ℤ[\overline{T}] is determined by the values c(0) and c(1), thus we will write c = [c(0), c(1)].

We define

\overline{a}_{t}(0) = ∑_{r even} |T_{r}| · a_{t}[r] = a_{t}[0] + 3 ∑_{m≥1} 2^{2m−1}a_{t}[2m],
\overline{a}_{t}(1) = ∑_{r odd} |T_{r}| · a_{t}[r] =  3 ∑_{m≥0} 2^{2m}a_{t}[2m+1],

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)] [ \begin{matrix}8 & 3 \\ −3 & −1 \end{matrix} ] = [\overline{a}_{t+1}(0), \overline{a}_{t+1}(1)],

[f_{m+2}, f_{m}] [ \begin{matrix}8 & 3 \\ −3 & −1 \end{matrix} ] = [f_{m+6}, f_{m+4}].

For the Fibonacci numbers, this is well-known:

[f_{m+6}, f_{m+4}] = [f_{m+5}, f_{m+4}] [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]
= [f_{m+1}, f_{m}] [ \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ]^{4} [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]
= [f_{m+2}, f_{m}] [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]^{−1} [ \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ]^{4} [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]

and the product of the matrices in the last line is just [ \begin{matrix}8 & 3 \\ −3 & −1 \end{matrix} ], 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

π : (T, E) → (\overline{T} , \overline{E}),

where π(x) = 0 if x is an even vertex of (T, E), and π(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 ∈ ℤ[T], define \overline{b} ∈ ℤ[\overline{T}] by

\overline{b}(i) = ∑_{x∈π^{−1}(i)} b(x) for i = 0, 1.

Observe that in this way we obtain from a_{t} ∈ ℤ[T] precisely \overline{a}_{t} ∈ ℤ[\overline{T}] as defined above.

On ℤ[\overline{T}], we also consider reflections, but we have to take into account the multiplicity of edges: There are the two reflections σ_{0}, σ_{1}, with

(σ_{0}c)(0) = −c(0) + 3c(1),  (σ_{0}c)(1) = c(1),
(σ_{1}c)(0) =  c(0),  (σ_{1}c)(1) = −c(1) + 3c(0),

for c ∈ ℤ[\overline{T}]. Note that this implies that

[c(0), c(1)] [\begin{matrix}8 & 3 \\ −3 & −1 \end{matrix}] = [(σ_{0}σ_{1}c)(0), (σ_{0}σ_{1}c)(1)].

We finish the proof by observing that \overline{a}_{t} = (σ_{0}σ_{1})^{t}\overline{a}_{0}. This follows directly from the following consideration: Let b ∈ ℤ[T], then

\overline{Φ_{0}b} = σ_{0}\overline{b}, \overline{Φ_{1}b} = σ_{1}\overline{b},

as it is easily verified.

4 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] and c_{t}[r] = a_{t+1}[2r + 1].

Then we obtain two “Pascal-like” triangles

[\begin{matrix}b_{0}[0] & b[1]_{0} & b_{0}[2] & ⋯ \\ b_{1}[0] & b_{1}[1] & b_{1}[2] & ⋯ \\ b_{2}[0] & b_{2}[1] & b_{2}[2] & ⋯ \\ b_{3}[0] & b_{3}[1] & b_{3}[2] & ⋯ \\ b_{4}[0] & b_{4}[1] & b_{4}[2] & ⋯ \\ & ⋯ \end{matrix} ] = [\begin{matrix}1 \\ 2 & 1 \\ 7 & 4 & 1 \\ 29 & 18 & 6 & 1 \\ 130 & 85 & 33 & 8 & 1 \\ & ⋯ \end{matrix} ]

and

[\begin{matrix}c_{0}[0] & c_{0}[1] & c_{0}[2] & ⋯ \\ c_{1}[0] & c_{1}[1] & c_{1}[2] & ⋯ \\ c_{2}[0] & c_{2}[1] & c_{2}[2] & ⋯ \\ c_{3}[0] & c_{3}[1] & c_{3}[2] & ⋯ \\ c_{4}[0] & c_{4}[1] & c_{4}[2] & ⋯ \\ & ⋯ \end{matrix} ] = [\begin{matrix}1 \\ 3 & 1 \\ 12 & 5 & 1 \\ 53 & 25 & 7 & 1 \\ 247 & 126 & 42 & 9 & 1 \\ & ⋯ \end{matrix} ]

The recursive definition is as follows: One starts with

b_{0}[r] = c_{0}[r] = δ_{r,0}

(using the Kronecker delta) and one continues for t ≥ 0 with

\begin{aligned}b_{t+1}[r] & = & c_{t}[r − 1] + 2c_{t}[r] − b_{t}[r], \\ c_{t+1}[r] & = & b_{t+1}[r] + 2b_{t+1}[r + 1] − c_{t}[r], \end{aligned}

where c_{t}[−1] = c_{t}[0].

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

f_{4t+2} = b_{t}[0] + 3 ∑_{r=1}^{t} 2^{2r−1} · b_{t}[r], f_{4(t+1)} = 3 ∑_{r=0}^{t} 2^{2r} · c_{t}[r].

5 Remarks

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 [10]. It is now recorded as sequence \underline{A132262}. There seems to be numerical evidence that the sequence a_{t}[1] is just the sequence \underline{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 [2] 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 [5].

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.

6 Motivation and background

Let us recall some basic concepts from the representation theory of quivers (see for example [1]); 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 α : a → b a k-linear map V_{α} : V_{a} → V_{b}. The function which assigns to the vertex a the k-dimension of V_{a} is called the dimension vector dim V of V; in case V_{a} = 0 for almost all vertices a, we obtain an element of ℤ[Q_{0}], where Q_{0} is the vertex set of Q, and 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 [7] has shown that the dimension vectors of the indecomposable objects are precisely the positive roots v of the Kac-Moody Lie-algebra defined by the underlying graph of the quiver. In case 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 → 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 [6] corresponding to the

3-Kronecker quiver is given by the generalized Cartan matrix [ \begin{matrix}2 & −3 \\ −3 & 2 \end{matrix} ], its positive real

roots are of the form

[f_{4t+2}, f_{4t}], and [f_{4t}, f_{4t+2}] with t ≥ 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

[ \begin{matrix}8 & 3 \\ −3 & −1 \end{matrix} ] 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 [ \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ] of the matrix

which exhibits the Fibonacci sequence, is conjugate to the matrix [ \begin{matrix}3 & 1 \\ −1 & 0 \end{matrix} ] 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 [3]. 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 [1]) 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 [9].

It remains to mention the covering theory of Gabriel and his students [4]. 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 ↦ \overline{V} preserves indecomposability and satisfies

dim \overline{V} = \overline{dim V} . (∗)

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 (∗).

7 Illustrations

The illustrations below show the functions a_{t} for t = 0, 1, 2 as well as the corresponding functions Φ_{1}a_{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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_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 1 1111111 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1} a_{0} Φ_{1}a_{0} a_{1} Φ_{1}a_{1} a_{2} Φ_{1}a_{2}

8

References

[1] M. Auslander, I. Reiten and S. Smalø, Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 1994.

[2] R. J. Baxter, Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

[3] I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, Coxeter functors and Gabriel’s theorem. Uspechi Mat. Nauk. 28 (1973), 19–33; Russian Math. Surveys 29 (1973), 17– 32.

[4] P. Gabriel, The universal cover of a representation-finite algebra. In Representations of Algebras, Puebla (1980), Lecture Notes in Mathematics, Vol. 903, Springer, 1981, pp. 68–105.

[5] Ph. Fahr, Infinite Gabriel-Roiter measures for the 3-Kronecker quiver. Dissertation Bielefeld, in preparation.

[6] V. G. Kac, Infinite-Dimensional Lie Algebras. 3rd edition, Cambridge University Press, 1990.

[7] V. G. Kac, Root systems, representations of quivers and invariant theory. Invariant theory, Montecatini (1982), Lecture Notes in Mathematics, Vol. 996, Springer, 1983, pp. 74–108.

[8] Th. Koshy, Fibonacci and Lucas Numbers with Applications. Wiley Series in Pure and Applied Mathematics, 2001.

[9] C. M. Ringel, Exceptional modules are tree modules. Linear Algebra and its applications (1998), 471–493.

[10] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, available electronically

at http://www.research.att.com/∼njas/sequences/.

2000 Mathematics Subject Classification: Primary 11B39, Secondary 16G20. Keywords: Fibonacci numbers, universal cover, 3-regular tree, 3-Kronecker quiver.

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

Received August 13 2007; revised version received January 22 2008. Published in Journal of Integer Sequences, February 9 2008.

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