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.