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

TL;DR

This paper introduces generalized $(p,q,t)$-Catalan numbers of Type A and B, expressing their expansions through permutation counting polynomials with descent statistics, using advanced combinatorial and bijective techniques.

Contribution

It extends Catalan number theory to new $(p,q,t)$-analogues for Types A and B, linking them to permutation pattern avoidance and descent statistics with novel combinatorial methods.

Findings

Type A $(p,q,t)$-Catalan numbers relate to permutations avoiding 321.

Type B $(p,q,t)$-Catalan numbers relate to permutations avoiding 3124, 4123, 3142, 4132.

Expansions are expressed via polynomials counting permutations with descent statistics.

Abstract

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 $\S_n(321)$ 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 $\S_n(3124, 4123, 3142, 4132)$ 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.