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_n2(321) = c_n2(123) and suggests that the n = 3 case may be tractable for arbitrary forbidden subsets of S_3.

References

Can we enumerate the matchings that avoid the set of patterns $P_1={123,132,213}$ and $P_{13}={132,213,321}$?

— Matchings and shape-Wilf-Equivalence of sets of patterns of length three I: Triples  (2609.08562 - Biswas et al., 8 Sep 2026) in Question 1 (label qn: leftover), Section “Conclusion and Open Problem”

What is $c_3k(\Lambda)$ for $\Lambda \subseteq _3$?

— Pattern avoidance in canon permutations  (2608.21351 - Laudone, 21 Aug 2026) in Question, Section 'Changing which parameter varies'