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.

On the sweep map for fuss rational Dyck paths Abstract   Our main contribution here is the discovery of a new family of standard Young tableaux      T    n    k      which are in bijection with the family      D    m  ,  n      of Rational Dyck paths for    m    k  xc3x97  n  xc2xb1  1    (the so called xe2x80x9cFussxe2x80x9d case). Using this family we give a new proof of the invertibility of the sweep map in the Fuss case by means of a very simple explicit algorithm. This new algorithm has running time    O  (  m    n  )   . It is independent of the ThomasWilliam algorithm.