Enumeration of pattern-avoiding 3-permutations and triangle bases

Determine the cardinality of the class of 3-permutations avoiding the patterns (12, 12) and (312, 231), equivalently the number of triangle bases of size n.

Background

The paper establishes a bijection between 3-permutations avoiding (12, 12) and (312, 231) and triangle bases of size n. Consequently, the two classes have identical cardinalities, but no exact enumeration is obtained. The authors note that the difficulty arises from the lack of a suitable recursive decomposition and provide only asymptotic upper and lower bounds.