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

TL;DR

This paper presents a new proof of Fine's identity by extending Wildberger's polynomial approach to include tubdigons, offering a combinatorial and algebraic perspective on the sum of multinomial coefficients.

Contribution

It generalizes Wildberger's polynomial formula to tubdigons and uses this to provide a novel proof of Fine's identity, connecting combinatorics and algebra.

Findings

New proof of Fine's identity using polynomial methods

Extension of Wildberger's polynomial formula to tubdigons

Demonstration of counting tubdigons via combinatorics and algebra

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.