Search for: All records

Abstract We give uniform formulae for the number of full reflection factorizations of a parabolic quasi‐Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus‐0 Hurwitz numbers. This paper is the culmination of a series of three. 

In 2019, Ceballos and Pons introduced the s-weak order on s-decreasing trees, for any weak composition s. They proved its lattice structure and conjectured that it could be realized as the 1-skeleton of a polyhedral subdivision of a zonotope of dimension n-1. We answer their conjecture in the case where s is a (strict) composition by providing three geometric realizations of the s-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third, obtained using tropical geometry, is the 1-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured. 

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron Pλ, and we give a formula for the dominant weight λ. Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of (3+1)-free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the ζ map from diagonal harmonics.