Classical pattern distributions in ${\mathcal{S}}_n(132)$ and ${\mathcal{S}}_n(123)$

Classical pattern avoidance and occurrence are well studied in the symmetric group ${\mathcal{S}}_n$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations.