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Hook length property of $$d$$-complete posets via $$q$$-integrals. (English) Zbl 1401.05035
Summary: 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 the others are verified by computer.

##### MSC:
 05A18 Partitions of sets 05A30 $$q$$-calculus and related topics 06A07 Combinatorics of partially ordered sets
SageMath
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