Conjectured Statistics for the Higher q;tCatalan Sequences This article describes conjectured combinatorial interpretations for the higher q,tCatalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund  and Haiman for the original q,tCatalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting t1 or q1 or t1q agree with the corresponding specializations of the GarsiaHaiman sequences. A third statistic occurs naturally in the combinatorial setting,  leading to the introduction of q,t,rCatalan sequences.  Similar combinatorial results are proved for these trivariate sequences.

The steepbounce zeta map in parabolic Cataland Abstract   As a classical object, the Tamari lattice has many generalizations, including xcexbdTamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to xcexbdTamari lattices for bounce paths xcexbd. We then introduce a new combinatorial object called xe2x80x9cleftaligned colorable treexe2x80x9d, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the SteepBounce Conjecture using a generalization of the famous zeta map in    q  ,  t   Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.