Fast Generation of Fibonacci Permutations

In 1985, Simion and Schmidt showed that |Sn(τ3)|, the cardinality of the set of all length n permutations avoiding the patterns τ3 = {123, 213, 132} is the Fibonacci numbers, fn+1. They also developed a constructive bijection between the set of all binary strings with no two consecutive ones and Sn(τ3). In May 2004, Egge and Mansour generalized this SimionSchmidt counting result and showed that, Sn(τp), the set of permutations avoiding the patterns τp = {12...p, 213, 132} is counted by the (p − 1)-generalized Fibonacci numbers, f (p−1) n+1 . The Simion-Schmidt’s set of binary strings is F (2) n−1, the well known Fibonacci strings, which is a special case of F (p) n , p-generalized Fibonacci strings having no p − 1 consecutive ones. In May 2001, Vajnovszki proposed a loopless algorithm for generating F (p) n , a Gray code for F (p) n . This algorithm has a constant worst case time while the Hamming distance between any two consecutive strings in F (p) n is one. In this paper we formalize and generalize the Simion-Schmidt bijection so that the new bijection now is between F (p−1) n−1 and Sn(τp), and we show that, Sn(τp), the image of the ordered list F (p) n through this generalized bijection is a list for all length n permutations avoiding the patterns τp = {12...p, 213, 132} with the Hamming distance between any two consecutive permutations bounded by (p−1), and so a Gray code for Sn(τp). We also propose a loopless algorithm, which is a modification of Vajnovszki’s algorithm, and which generates Sn(τp) also in constant worst case time.