A positional statistic for 1324-avoiding permutations

We consider the class Sn(1324) of permutations of size n that avoid the pattern 1324 and examine the subset S_n^a≺ⁿ(1324) of elements for which a ≺ n ≺ [a − 1], a ≥ 1. This notation means that, when written in one-line notation, such a permutation must have a to the left of n, and the elements of {1, . . ., a − 1} must all be to the right of n. For n ≥ 2, we establish a connection between the subset of permutations in S_n¹≺ⁿ(1324) having the 1 adjacent to the n (called primitives), and the set of 1324-avoiding dominoes with n − 2 points. For a ∈ {1, 2}, we introduce constructive algorithms and give formulas for the enumeration of S_n^a≺ⁿ(1324) by the position of a relative to the position of n. For a ≥ 3, we formulate some conjectures for the corresponding generating functions.