Hook length property of $d$-complete posets via $q$-integrals

TL;DR

This paper presents a new proof of the hook length formula for $d$-complete posets using $q$-integrals, involving case-by-case analysis and computational verification.

Contribution

It introduces a novel proof method for the hook length formula employing $q$-integrals and detailed case analysis.

Findings

Successful evaluation of multiple $q$-integrals related to $d$-complete posets

Verification of some $q$-integrals through computer-assisted calculations

Reinforcement of the hook length formula's validity via new proof technique

Abstract

The hook length formula for $d$-complete posets states that the $P$-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using $q$-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the $P$-partition generating function for each case as a $q$-integral and then we evaluate the $q$-integrals. Several $q$-integrals are evaluated using partial fraction expansion identities and others are verified by computer.