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

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.