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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.

05A18 Partitions of sets
05A30 \(q\)-calculus and related topics
06A07 Combinatorics of partially ordered sets
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[1] Andrews, G. E.; Askey, R.; Roy, R., Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71, (1999)
[2] Frame, J. S.; Robinson, G.d. B.; Thrall, R. M., The hook graphs of the symmetric groups, Canad. J. Math., 6, 316-324, (1954) · Zbl 0055.25404
[3] Ishikawa, M.; Tagawa, H., Schur function identities and hook length posets, (19th International Conference on Formal Power Series and Algebraic Combinatorics, (2007), Nankai University: Nankai University Tianjin, China)
[4] Ishikawa, M.; Tagawa, H., Leaf posets and multivariate hook length property, RIMS Kôkyûroku, 1913, 67-80, (2014)
[5] Kim, J. S.; Stanton, D., On q-integrals over order polytopes, Adv. Math., 308, 1269-1317, (2017) · Zbl 1355.05051
[6] Milne, S., A q-analog of the Gauss summation theorem for hypergeometric series in \(\operatorname{U}(n)\), Adv. Math., 72, 1, 59-131, (1988) · Zbl 0658.33005
[7] Morales, A.; Pak, I.; Panova, G., Hook formulas for skew shapes I. q-analogues and bijections, J. Combin. Theory Ser. A, 154, 350-405, (2018) · Zbl 1373.05026
[8] Nakada, K., q-hook formula of Gansner type for a generalized Young diagram, (21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., AK, (2009), Assoc. Discrete Math. Theor. Comput. Sci.: Assoc. Discrete Math. Theor. Comput. Sci. Nancy), 685-696 · Zbl 1392.05117
[9] Naruse, H., Schubert calculus and hook formula, (2014), Talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, available at
[10] Naruse, H.; Okada, S., Skew hook formula for d-complete posets
[11] Proctor, R. A., Dynkin diagram classification of λ-minuscule Bruhat lattices and of d-complete posets, J. Algebraic Combin., 9, 1, 61-94, (1999) · Zbl 0920.06003
[12] Proctor, R. A., Minuscule elements of Weyl groups, the numbers game, and d-complete posets, J. Algebra, 213, 1, 272-303, (1999) · Zbl 0969.05068
[13] Proctor, R. A., d-complete posets generalize Young diagrams for the hook product formula: partial presentation of proof, RIMS Kôkyûroku, 1913, 120-140, (2014)
[14] Proctor, R. A.; Scoppetta, L. M., d-complete posets: local structural axioms, properties, and equivalent definitions
[15] Rainville, E. D., Special Functions, (1971), Chelsea Publishing Co.: Chelsea Publishing Co. Bronx, N.Y. · Zbl 0231.33001
[16] Rosengren, H., Elliptic hypergeometric series on root systems, Adv. Math., 181, 2, 417-447, (2004) · Zbl 1066.33017
[17] Stanley, R. P., Enumerative Combinatorics, vol. 1, (2011), Cambridge University Press: Cambridge University Press New York/Cambridge · Zbl 1247.05003
[18] (2017), The Sage Developers. SageMath, the Sage Mathematics Software System (Version 7.5.1)
[19] Warnaar, S. O., q-Selberg integrals and Macdonald polynomials, Ramanujan J., 10, 2, 237-268, (2005) · Zbl 1086.33018
[20] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis, (1927), Cambridge University Press: Cambridge University Press Cambridge, (reprinted 1996) · JFM 53.0180.04
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