A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

Wenjie Fang (1) , Henri Mühle (2) , Jean-Christophe Novelli (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 TU Dresden - Technische Universität Dresden = Dresden University of Technology

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.

Domaines: Combinatoire [math.CO]

Wenjie Fang, Henri Mühle, Jean-Christophe Novelli. A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group. The Electronic Journal of Combinatorics, 2021, 28 (3), ⟨10.37236/10578⟩. ⟨hal-03667557⟩