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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
Software:
SageMath
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References:
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