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

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 of d-complete posets using q-integrals. Proctor proved that any connected d-complete poset can be uniquely decomposed into irreducible d-complete posets and classified all irreducible d-complete posets. In this work, we prove the hook length property of all the irreducible d-complete posets. 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.