A multivariate "inv" hook formula for forests

TL;DR

This paper extends the inversion number hook formula for forests to a multivariate setting, motivated by modular invariant theory of finite general linear groups, building on previous q-generalizations involving major index and inversion statistics.

Contribution

It introduces a multivariate generalization of the inversion number hook formula for forests, connecting combinatorics with modular invariant theory.

Findings

Proves a multivariate inversion number hook formula for forests.

Links combinatorial formulas to modular invariant theory.

Builds on previous q-generalizations involving major index and inversion statistics.

Abstract

Bjoerner and Wachs provided two q-generalizations of Knuth's hook formula counting linear extensions of forests: one involving the major index statistic, and one involving the inversion number statistic. We prove a multivariate generalization of their inversion number result, motivated by specializations related to the modular invariant theory of finite general linear groups.