We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.
@article{10_37236_7844,
author = {Henri M\"uhle and Nathan Williams},
title = {Tamari lattices for parabolic quotients of the symmetric group},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {4},
doi = {10.37236/7844},
zbl = {1453.20051},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7844/}
}
TY - JOUR
AU - Henri Mühle
AU - Nathan Williams
TI - Tamari lattices for parabolic quotients of the symmetric group
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/7844/
DO - 10.37236/7844
ID - 10_37236_7844
ER -
%0 Journal Article
%A Henri Mühle
%A Nathan Williams
%T Tamari lattices for parabolic quotients of the symmetric group
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/7844/
%R 10.37236/7844
%F 10_37236_7844
Henri Mühle; Nathan Williams. Tamari lattices for parabolic quotients of the symmetric group. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/7844
