Arrow-Wilf equivalence of (12; 3→3) and (23; 1→1)

Establish whether the arrow patterns (12; 3→3) and (23; 1→1) are arrow-Wilf-equivalent, and, if so, construct a direct bijection or provide another proof of the equivalence.

Background

The paper derives explicit avoidance formulas for both (12; 3→3) and (23; 1→1). The resulting sequences agree for every value of n tested, suggesting an underlying arrow-Wilf equivalence.

Despite this numerical agreement, the formulas have substantially different forms, and the authors report that they were unable to prove the equivalence by a direct bijection or any other method. The question therefore remains explicitly open.

References

We have also established several arrow-Wilf equivalences, many of which arise from reversal, complementation, and insertion operations. However, a number of these equivalences were obtained through ad hoc arguments, suggesting that a more unified framework for arrow-Wilf equivalence may exist. We leave this as an open problem for future investigation.

— Arrow-Wilf equivalences and enumerative results for short arrow patterns  (2609.29392 - Zhou et al., 24 Sep 2026) in Section 6, Final remarks and future directions