Emil Verkama: Counting 1324-avoiders with few inversions

ABSTRACT: 

Little is known of 1324-avoiding permutations. Whereas the sets of permutations avoiding any single pattern of length four other than 1324 or 4231 were enumerated in the 1990s, not even the exponential growth rate of the 1324-avoiders has been determined. Using a new notion of almost-decomposability, we present a structural characterization of 1324-avoiding n-permutations with at most 2n-7 inversions. This result allows us to enumerate those permutations, and it partially resolves a conjecture of Claesson, Jelínek and Steingrímsson, according to which the 1324-avoiding permutations are inversion-monotone. A proof of the full conjecture would improve the best known upper bound for the exponential growth rate.

This talk is based on ongoing work with Svante Linusson.