Abstract
In the theory of Clebsch-Gordan coefficients, one may recognize the domain space as the set of weakly semi-magic squares of size three. Two partitions on this set are considered: a triangle-hexagon model based on top lines, and one based on the orbits under a finite group action. In addition to giving another proof of McMahon’s formula, we give a generating function that counts the so-called trivial zeros of Clebsch-Gordan coefficients and its associated quasi-polynomial.
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References
Beck, M., Zaslavsky, T.: Six little squares and how their numbers grow. J. Integer Seq. 13, 1–45 (2010)
Bóna, M.: A new proof of the formula for the number of \(3\times 3\) magic squares. Math. Mag. 70, 201–203 (1997)
Donley, R.W., Jr., Kim, W.G.: A rational theory of Clebsch-Gordan coefficients. In: Representation theory and harmonic analysis on symmetric spaces. American Mathematical Society, Providence. Contemp. Math., Vol. 714, 115–130 (2018)
Donley, R. W., Jr.: Central values of Clebsch-Gordan coefficients. In: Combinatorial and additive number theory III. Springer, New York. Springer Proc. in Math. and Stat., Vol. 297, 26 pp. (2019)
Donley, R. W., Jr.: Binomial arrays and generalized Vandermonde identities. Preprint, 28 pp (2019)
Louck, J. D.: Applications of unitary symmetry and combinatorics. World Scientific, Teaneck, N.J. (2011)
MacMahon, P. A.: Combinatory Analysis, Vols. 1 and 2. Cambridge University Press, (1916); reprinted by Chelsea, New York (1960) and Dover, New York (2004)
Rao, K. S., Van der Jeugt, J., Raynal, J., Jagannathan, R., Rajeswari, V.: Group theoretic basis for the terminating \({}_3F_2(1)\) series. J. Phys. A 25, 861–876 (1992)
Raynal, J., Van der Jeugt, J., Rao, K.S., Rajeswari, V.: On the zeros of 3j coefficients: Polynomial degree versus recurrence order. J. Phys. A 26, 2607–2623 (1993)
Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org/.
Stanley, R.: Combinatorics and commutative algebra. Second edition. Birkhäuser, Boston (1996)
Stanley, R.: Enumerative combinatorics, Vol. 1. Second edition. Cambridge University Press, New York (2011)
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Donley, R.W. (2021). Partitions for Semi-magic Squares of Size Three. In: Nathanson, M.B. (eds) Combinatorial and Additive Number Theory IV. CANT 2020. Springer Proceedings in Mathematics & Statistics, vol 347. Springer, Cham. https://doi.org/10.1007/978-3-030-67996-5_7
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