Enumeration of canon permutations avoiding two length-three patterns for n = 3

Determine c_3^k(Λ) for every subset Λ of the length-three permutation patterns S_3.

Background

Most of the paper fixes k and varies n, whereas this question reverses the roles by fixing n = 3 and allowing k to vary. The question is motivated by conjectured formulas whose counts are simple when n is fixed.

The paper specifically highlights the still-unresolved enumeration of c_n^2(321) = c_n^2(123) and suggests that the n = 3 case may be tractable for arbitrary forbidden subsets of S_3.