Combinatorial isomorphism between Fibonacci classes

Abstract

In 1985 Simion and Schmidt showed that the set S n (T 3) of length n permutations avoiding the set of patterns T 3={123, 132, 213} is counted by (the second order) Fibonacci numbers. They also presented a constructive bijection between the set F n–1 of length (n–1) binary strings with no two consecutive 1s and S n (T 3). In 2005, Egge and Mansour generalized the first Simion-Simion’s result and showed that S n (T p ), the set of permutations avoiding the patterns T p ={12…p, 132, 213}, is counted by the (p–1)th order Fibonacci numbers. In this paper we extend the second Simion-Schmidt’s result by giving a bijection between the set of length (n–1) binary strings with no (p–1) consecutive 1s, and the set S n (T p ). Moreover, we show that this bijection is a combinatorial isomorphism, i.e., a closeness-preserving bijection, by which we transform a known Gray code (or equivalently Hamiltonian path) and exhaustive generating algorithm for into similar results for S n (T p ).