An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients

TL;DR

This paper establishes a new combinatorial identity linking $n$-nacci numbers with partitions and binomial coefficients, extending classical Fibonacci identities and analyzing related partial orders.

Contribution

It introduces a novel identity connecting $n$-nacci numbers to partitions and binomial products, generalizing Fibonacci identities and exploring partition orderings.

Findings

Derived an identity expressing $n$-nacci numbers as sums over partitions.

Extended classical Fibonacci identities to $n$-nacci numbers.

Compared partial orders on partitions induced by final types.

Abstract

We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci numbers as sums of products of binomial coefficients over these partitions, generalizing the classical identity for $n = 2$ that expresses Fibonacci numbers in this way. We also examine how the partial order on the set of all partitions of a fixed integer induced by the ordering of final types compares with two natural partial orders on the same set.