On the Number of Permutations Avoiding a Given Pattern

Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 < . . . < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = |{τ ∈ Sn| τ avoids σ}|. Stanley and Wilf conjectured that for any σ ∈ Sk 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 ∈ Sk, F (n, σ) ≤ cnγ (n), where c = c(σ) and γ∗(n) is an extremely slow growing function, related to the Ackermann hierarchy.