  Zeckendorf proved that every integer can be written uniquely as a sum of
non-consecutive Fibonacci numbers $\{F_n\}$, and later researchers showed that
the distribution of the number of summands needed for such decompositions of
integers in $[F_n, F_{n+1})$ converges to a Gaussian as $n\to\infty$.
Decomposition problems have been studied extensively for a variety of different
sequences and notions of a legal decompositions; for the Fibonacci numbers, a
legal decomposition is one for which each summand is used at most once and no
two consecutive summands may be chosen. Recently, Chen et al. [CCGJMSY]
generalized earlier work to $d$-dimensional lattices of positive integers;
there, a legal decomposition is a path such that every point chosen had each
component strictly less than the component of the previous chosen point in the
path. They were able to prove Gaussianity results despite the lack of
uniqueness of the decompositions; however, their results should hold in the
more general case where some components are identical. The strictly decreasing
assumption was needed in that work to obtain simple, closed form combinatorial
expressions, which could then be well approximated and led to the limiting
behavior. In this work we remove that assumption through inclusion-exclusion
arguments. These lead to more involved combinatorial sums; using generating
functions and recurrence relations we obtain tractable forms in $2$ dimensions
and prove Gaussianity again; a more involved analysis should work in higher
dimensions.
