Rectangular tileability and complementary tileability are undecidable

Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.Comment: 16 pages, 8 figure

Pattern avoidance is not P-recursive

Let $F \subset S_k$ be a finite set of permutations and let $C_n(F)$ denote the number of permutations $\sigma$ in $S_n$ avoiding the set of patterns $F$. The Noonan-Zeilberger conjecture states that the sequence ${C_n(F)}$ is P-recursive. We use Computability Theory to disprove this conjecture.Comment: 19 page

Complexity problems in enumerative combinatorics

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.Comment: 31 pages; an expanded version of the ICM 2018 paper (Section 4 added, refs expanded