$(p,q,t)$-Catalan continued fractions, gamma expansions and pattern avoidances

We introduce a kind of $(p,q,t)$-Catalan numbers of Type A by generalizing the Jacobian type continued fraction formula, we proved that the corresponding expansions could be expressed by the polynomials counting permutations on ${\mathrm{\S }}_n(321)$ by various descent statistics. Moreover, we introduce a kind of $(p,q,t)$-Catalan numbers of Type B by generalizing the Jacobian type continued fraction formula, we proved that the Taylor coefficients and their $\gamma$-coefficients could be expressed by the polynomials counting permutations on ${\mathrm{\S }}_n(3124,4123,3142,4132)$ by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.