From Fibonacci Numbers to Central Limit Type Theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. This has been generalized to the following: given nonnegative integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1} =c_1H_n+c_2H_{n-1}+...+c_nH_1+1$ $(1\le n< L)$ and $H_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L}$ $(n\geq L)$, every positive integer can be written uniquely as $\sum a_iH_i$ under natural constraints on the $a_i$'s, the mean and the variance of the numbers of summands for integers in $[H_{n}, H_{n+1})$ are of size $n$, and the distribution of the numbers of summands converges to a Gaussian as $n$ goes to the infinity. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to a multitude of other problems (in the sequel paper \cite{BM} we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely $-(21-2\phi)/(29+2\phi) \approx -0.551058$.Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the number of summands in Zeckendorf decompositions. Version 2.0 (mostly correcting missing references to previous literature

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let \UT_n(\FF_q) denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of \FF_q. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of \UT_{n+1}(\FF_q) is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of \UT_n(\FF_q) is a polynomial in $q-1$ whose coefficients are Delannoy numbers and whose values give a $q$-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in $q-1$ with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor corrections, final versio