Restricted Permutations Related to Fibonacci Numbers and k-Generalized Fibonacci Numbers

A permutation π ∈ Sn is said to avoid a permutation σ ∈ Sk whenever π contains no subsequence with all of the same pairwise comparisons as σ. In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 123, 132, and 213 is the Fibonacci number Fn+1. In this paper we generalize this result in two ways. We first show that the number of permutations which avoid 132, 213, and 12 . . . k is the k − 1-generalized Fibonacci number Fk−1,n+1. We then show that the number of permutations which avoid 123, 132, and k−1 k−2 . . . 21k is also the k−1-generalized Fibonacci number Fk−1,n+1. We go on to show that the number of permutations in Sn which avoid 132, k k − 1 . . . 4213, and 2341 is given by a polynomial plus a linear combination of two Fibonacci numbers. We give explicit enumerations for k ≤ 6. We begin to generalize this result by showing that the number of permutations in Sn which avoid 132, 213, and 23 . . . k1 is ∑n i=1 Fk−2,i. We conclude with several conjectures and open problems.