On Summand Minimality of Generalized Zeckendorf Decompositions

Zeckendorf proved that every number can be uniquely represented as a sum of non-consecutive Fibonacci numbers. This has been extended in many ways, including to linear recurrences $H_n=c_1 H_{n-1} + \cdots + c_t H_{n-t}$ where the $c_i$ are non-negative integers and $c_1$, $c_t \ge 1$. Every number has a unique generalized Zeckendorf decomposition (gzd) -- a representation composed of blocks that are lexicographically less than $(c_1,\dots,c_t)$, which we call the signature. We prove that the gzd of a positive integer $m$ uses the fewest number of summands out of all representations for $m$ using the same recurrence sequence, for all $m$, if and only if the signature of the linear recurrence is weakly decreasing (i.e., $c_1 \ge \cdots \ge c_t$). Following the parallel with well-known base $d$ representations, we develop a framework for naturally moving between representations of the same number using a linear recurrence, which we then utilize to construct an algorithm to turn any representation of a number into the gzd. To prove sufficiency, we show that if the signature is weakly decreasing then our algorithm results in fewer summands. To prove necessity we proceed by divide and conquer, breaking the analysis into several cases. When $c_1 > 1$, we give an example of a non-gzd representation of a number and show that it has fewer summands than the gzd by performing the same above-mentioned algorithm. When $c_1 = 1$, we non-constructively prove the existence of a counterexample by utilizing the irreducibility of a certain family of polynomials together with growth rate arguments.