A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

Abstract

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.

26 pages, 2 tables, 9 figures. Comments are welcome. In version 2, we added a new section concerning Tamari lattices via Dyck paths, and show that our bijection is related to a well-known map exchanging two sequence statistics

References in corpus (1)

• The Steep-Bounce Zeta Map in Parabolic Cataland

Cited by in corpus (1)

• Parabolic Tamari Lattices in Linear Type B