A New Proof of Fine's Identity using Wildberger's Polynomial Formula

Abstract

In 1959, N. J. Fine showed that the sum of the multinomial coefficients corresponding to the partitions of a natural number $n$ into $r$ parts is a binomial coefficient: $$ \sum_{\substack{k_1 + k_2 + k_3 + {}\ldots = r \\ k_1 + 2k_2 + 3k_3 + {}\ldots = n }} \binom{r}{k_1, k_2, k_3, \ldots } = \binom{n - 1}{r - 1} $$ Fine gives a rather pithy proof, though we're still stuck on the part that says, ``We begin with an important though obvious remark.'' In 2025, Wildberger and Rubine gave the series solution to the general polynomial, derived from a non-associative algebra of roofed, subdivided polygons they call \textit{subdigons}. We generalize subdigons to \textit{tubdigons}, which include 2-gons, and count tubdigons of a given type two ways: through a simple counting argument (backed up by the combinatorics literature) and by using Wildberger's polynomial formula to solve the polynomial implied by the multiset specification of tubdigons. Comparing corresponding terms yields Fine's Identity.