Abstract
The q-identities corresponding to Sylvester’s bijection between odd and strict partitions are investigated. In particular, we show that Sylvester’s bijection implies the Rogers-Fine identity and give a simple proof of a partition theorem of Fine, which does not follow directly from Sylvester’s bijection. Finally, the so-called (m, c)-analogues of Sylvester’s bijection are also discussed.
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References
K. Alladi, “A fundamental invariant in the theory of partitions, Topics in Number Theory, (University Park, PA, 1997), 101–113, Math. Appl. 467 Kluwer Acad. Publ., Dordrecht 1999.
G. Andrews, “On basic hypergeometric seris, mock-theta functions, and partitions (II),” Quart. J. Math. 17(2) (1966), 132–143.
G. Andrews, “Two theorems of Gauss and allied identities proved arithmetically,” Pacific J. Math. 41 (1972), 563–568.
G. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Compagny, 2nd printing, with revision, 1981.
G. Andrews, “On a partition theorem of N. Fine,” J. Nat. Acad. Math. 1 (1983), 105–107.
G. Andrews, “Use and extension of Frobenius representation of partitions,” in Enumeration and Design, (D.M. Jackson and S. A. Vanston, (eds.)), Academic Press, New York, 1984, pp. 51–65.
C. Bessenrodt, “A bijection for Lebesgue’s partition identity in the spirit of Sylvester,” Discrete Math. 132 (1994), 1–10.
M. Bousquet-Mélou and K. Eriksson, “Lecture Hall Partitions,” The Ramanujan Journal, 1:(101–111)/2:165–185, (1997).
D. Bressoud, Proofs and Confirmations, The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press, 1999.
N. Fine, “Some new results on partitions,” in Proc. Nat. Acad. Sci. U.S.A. 34 (1948).
N. Fine, “Basic hypergeometric series and applications,” Amer. Math. Soc., Math. Surveys and Monographs (27) 1988.
G. Gasper and M. Rahman, “Basic Hypergeometric Series,” Encyclopedia of Math. and Its Applications 35, Cambridge Univ. Press, 1990.
J.W.L. Glaisher, “A theorem in partitions,” Messenger of Math. 12 (1883), 1111–1112.
M.D. Hirschhorn, “Sylvester’s partition theorem and a related result,” Michigan J. Math. 21 (1974), 133–136.
D.S. Kim and A.Y. Yee, “A note on the partitions into odd and distinct parts,” The Ramanujan J. 2(3) (1999), 165–185.
A. Lascoux, “Sylvester’s bijection between strict and odd partitions,” Discrete Math. 277(1–3) (2004), 275–278.
V.A. Lebesgue, “Sommation de quelques séries,” J. Math. Pures Appl. 5 (1840), 42–71.
P.A. Macmahon, Combinatory Analysis, Chelsea Reprints (1984).
I. Pak and A. Postnikov, “A generalization of Sylvester’s identity,” Discrete Math. 178 (1998), 277–281.
J.J. Sylvester, “A constructive theory of partitions, arranged in three acts, an interact and an exodion,” Amer. J. Math. 5 (1882), 251–330.
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2000 Mathematics Subject Classification: Primary—05A17, 05A15, 33D15, 11P83
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Zeng, J. The q-Variations of Sylvester’s Bijection Between Odd and Strict Partitions. Ramanujan J 9, 289–303 (2005). https://doi.org/10.1007/s11139-005-1869-2
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DOI: https://doi.org/10.1007/s11139-005-1869-2