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

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