A positional statistic for 1324-avoiding permutations

We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. 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 $\{1,\dots ,a-1\}$ must all be to the right of $n$. For $n\ge 2$, we establish a connection between the subset of permutations in $S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives), and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in \{1,2\}$, we introduce constructive algorithms and give formulas for the enumeration of $S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$. For $a\ge 3$, we formulate some conjectures for the corresponding generating functions.