Emil Verkama: Inversions and 1324, again

Abstract: I will discuss some recent progress on the inversion-monotonicity conjecture for pattern-avoiding permutations. A pattern p is called inversion-monotone if the number of p-avoiding permutations with a fixed number of inversions is nondecreasing in n. This property is conjectured to hold for every pattern except the identity. In joint work with Linusson we attacked the notorious pattern p = 1324, the first nontrivial case, and showed that this instance of the conjecture is at least 50% true.

In ongoing work with Claesson, Linusson and Ulfarsson we tackle the inversion-monotonicity of 1324 from a different angle: can we find inversion-monotone sets that contain 1324? Our results include combinatorial proofs of the inversion-monotonicity of {1324, 231} and {1324, 2314, 3214, 4213}, as well as a complete analysis of the limiting behaviour of {1324, p}-avoiding permutations with a fixed number of inversions, where p is any pattern of length 4.