On the Number of Permutations Avoiding a Given Pattern

Let σ∈S_k and τ∈S_n be permutations. We say τ contains σ if there exist 1≤x₁<x₂<...<x_k≤n such that τ(x_i)<τ(x_j) if and only if σ(i)<σ(j). If τ does not contain σ we say τ avoids σ. Let F(n, σ)=(τ∈S_nτavoidsσ). Stanley and Wilf conjectured that for any σ∈S_k there exists a constant c=c(σ) such that F(n, σ)≤cⁿ for all n. Here we prove the following weaker statement: For every fixed σ∈S_k, F(n, σ)≤c^nγ*(n), where c=c(σ) and γ*(n) is an extremely slow growing function, related to the Ackermann hierarchy.