Enumeration of nonnesting canon permutations avoiding 321

Determine the enumeration of canon permutations avoiding 321 in the case k = 2, equivalently determine c_n^2(321) = c_n^2(123).

Background

The paper notes that much prior work fixes k = 2 and varies n. In that setting, the enumeration of canon permutations avoiding 321 is identified with the equivalent avoidance problem for 123 by symmetry.

Unlike several neighboring cases treated in the paper, this enumeration is explicitly described as still open.