Enumeration of the remaining arrow pattern (32; 1→3)

Determine the avoidance enumeration of the arrow pattern (32; 1→3), the only unresolved arrow pattern of the form (ν; b→c) with |ν|≤2 remaining after the results of Archer and Laudone and the present paper.

Background

The paper studies arrow patterns, which simultaneously impose conditions on the one-line notation of a permutation and on the cycle structure obtained through Foata’s transformation. It enumerates all relevant size-three cases except (32; 1→3).

The unresolved pattern is explicitly identified as the sole remaining case with underlying pattern length at most two. Determining its avoidance sequence would complete the enumeration program for this range of arrow patterns.