Classical Fibonacci compositions

Abstract

There are three long-known types of restricted integer compositions whose counts match the Fibonacci sequence:\ one from ancient India and two from 19th century England. We give proofs of these enumeration results using tiling arguments and discuss how these can be used in combinatorial proofs of several Fibonacci number identities. With MacMahon's notion of conjugation, we show that, for every $n \ge 2$, the compositions of $n$ include subsets whose sizes satisfy the Fibonacci recurrence.