Permutation patterns are hard to count

Abstract

Let F ⊂ Sk be a finite set of permutations and let Cn(F) denote the number of permutations σ σ Sn avoiding the set of patterns F. We prove that {Cn(F)} cannot be computed in time polynomial in n, unless EXP = ⊕EXP. Our tools also allow us to disprove the Noonan-Zeilberger conjecture which states that the sequence {Cn(F)} is P-recursive.