# Multivariate polynomials for generalized permutohedra

## TL;DR

This paper introduces a new $q$-analogue of the $h$-polynomial for generalized permutohedra using Mahonian statistics on acyclic posets, with explicit computations for various examples including associahedra.

## Contribution

It develops a novel $q$-analogue framework for the $h$-polynomial of generalized permutohedra, especially nestohedra, and provides explicit calculations for key examples.

## Key findings

- New $q$-analogue of the $h$-polynomial for generalized permutohedra
- Explicit computations for S_n-invariant nestohedra and graph associahedra
- Alternative $q$-analogue to Narayana numbers for associahedra

## Abstract

Using the notion of Mahonian statistic on acyclic posets, we introduce a $q$-analogue of the $h$-polynomial of a simple generalized permutohedron. We focus primarily on the case of nestohedra and on explicit computations for many interesting examples, such as $S_n$-invariant nestohedra, graph associahedra, and Stanley--Pitman polytopes. For the usual (Stasheff) associahedron, our generalization yields an alternative $q$-analogue to the well-studied Narayana numbers.