Five subsets of permutations enumerated as weak sorting permutations

In this paper, we show that the number of members of Sn avoiding any one
of five specific triples of 4-letter patterns is given by sequence A111279, which is known to count weak sorting permutations. By numerical evidence, there are no other (nontrivial) triples of 4-letter patterns giving rise to this sequence. We make use of a variety of methods [5, 6] in proving our result, including recurrences, the kernel method, direct counting, and bijections.