Permutation patterns are hard to count

Abstract

Let F ⊂ S k be a finite set of permutations and let Cn(F ) denote the number of permutations 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.

Complexity Problems in Enumerative Combinatorics

“…We extend this result in a forthcoming paper [67], where we construct a D-transcendent sequence {A n (F)}, for some F ⊂ S 80 . Both proofs involve embedding of Turing machines into the problem modulo 2.…”

“…It would be interesting to see if this sequence is D-algebraic. In a forthcoming paper [67] we construct an explicit but highly artificial non-symmetric set S ⊂ F k × F ℓ with D-transcendental cogrowth sequence. In [95] we use the tools in [94] to prove that groups have an uncountable set of spectral radii…”

The number of occurrences of patterns in a random tree or forest permutation

“…(See e.g. Garrabrant and Pak [4] for some related impossibility results supporting this.) Therefore, typically these classes are studied one by one, with methods depending on the knowledge of some structure theorem for permutations in that particular class.…”

Finitely generated subgroups of free groups as formal languages and their cogrowth

“…The cogrowth series (6.1) represents a function of complex variable z ∈ C analytic at disc around z = 0 of radius R ≥ 1 2m − 1 . The major questions are: Under what conditions H(z) is rational, algebraic and belongs to distinguished class of analytic functions, like for instance the class of D-finite functions studied in [6].…”