On a recurrence related to 321-avoiding permutations

Dokos et al. recently conjectured that the distribution polynomial $f_n(q)$ on the set of permutations of size $n$ avoiding the pattern $321$ for the number of inversions is given by $$f_n(q)=f_{n-1}(q)+\sum_{k=0}^{n-2}q^{k+1}f_k(q)f_{n-1-k}(q), \qquad n \geq 1,$$ with $f_0(q)=1$, which was later proven in the affirmative, see \cite{CEKS}. In this note, we provide a new proof of this conjecture, based on the scanning-elements algorithm described in \cite{FM}, and present an identity obtained by equating two explicit formulas for the generating function $\sum_{n\geq0}a_n(q)x^n$.