Enumeration of canon permutations avoiding (12^j1) in the intermediate regime

Determine the number c_n^k((12^j1)) of canon permutations avoiding the symmetric pattern set (12^j1) = {12^j1, 21^j2} for n ≥ 3 and 4 ≤ j < k < 2(j−1), including the special case k = j+1.

Background

The paper completely enumerates avoidance of (12^j1) when k > 2(j−1), when k = 2(j−1), and when j ≥ k. For n = 2, it also derives an explicit formula using standard skew tableaux and Dyck paths. The remaining intermediate range j < k < 2(j−1) is substantially more difficult because the skew-tableau reduction used for n = 2 does not capture all constraints once n ≥ 3.

The smallest unresolved instance is j = 4 and k = 5. The paper lists the beginning of the corresponding lattice-word enumeration as 1, 19, 795, 48875, 3780362, 340746821, 34264303289, …, with the canon-permutation count obtained by multiplying by n!.