Abstract
In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.
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References
Andrews, G.: The theory of Partitions. Encyclopedia of Mathematics and Its Applications, vol. II. Addison-Wesley, Reading (1976)
Andrews, G., Eriksson, K.: Integer Partitions. Cambridge University Press, Cambridge (2004)
Baruah, N.D., Berndt, B.C.: Partition identities and Ramanujan’s modular equations. J. Comb. Theory, Ser. A 114(6), 1024–1045 (2007)
Berndt, B.C.: “Ramanujan’s Notebooks”, Part III. Springer, New York (1991)
Berndt, B.C.: Partition-theoretic interpretations of certain modular equations of Schröter, Russell, and Ramanujan. Ann. Comb. 11(2), 115–125 (2007)
Farkas, H.M., Kra, I.: Partitions and theta constant identities. In: The Mathematics of Leon Ehrenpreis. Contemp. Math., vol. 251, pp. 197–203. Amer. Math. Soc., Providence (2000)
Kim, S.: Bijective proofs of partition identities arising from modular equations. J. Comb. Theory, Ser. A 116(3), 699–712 (2009)
Pak, I.: Partition bijections, a survey. Ramanujuan J. 12, 5–75 (2006)
Ramanujan, S.: Notebooks, vol. 1–2. Tata Institute of Fundamental Research, Bombay (1957)
Russell, R.: On κλ−κ′λ′ modular equations. Proc. Lond. Math. Soc. 19, 90–111 (1887)
Russell, R.: On modular equations. Proc. Lond. Math. Soc. 21, 351–395 (1890)
Sandon, C., Zanello, F.: Warnaar’s bijection and colored partition identities, I. J. Comb. Theory, Ser. A 120(1), 28–38 (2013)
Schröter, H.: Beiträge zur Theorie der elliptischen Funktionen. Acta Math. 5, 205–208 (1884)
Warnaar, S.O.: A generalization of the Farkas and Kra partition theorem for modulus 7. J. Comb. Theory, Ser. A 110(1), 43–52 (2005)
Acknowledgements
This work, along with the previous paper [12], is the result of the first author’s MIT senior thesis, done in Summer and Fall 2011 under the supervision of the second author, and funded by the Institute through two UROP grants. We wish to thank Nayandeep Deka Baruah for pointing us to Theorem 8.1 of [3], and the anonymous referee for a careful reading of our manuscript and helpful comments. The second author warmly thanks Richard Stanley for his terrific hospitality during calendar year 2011, the MIT Math Department for partial financial support, and Dr. Gockenbach and the Michigan Tech Math Department, from which he was on partial leave, for extra Summer support.
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Sandon, C., Zanello, F. Warnaar’s bijection and colored partition identities, II. Ramanujan J 33, 83–120 (2014). https://doi.org/10.1007/s11139-013-9465-3
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DOI: https://doi.org/10.1007/s11139-013-9465-3
Keywords
- Partition identity
- Colored partition
- Farkas–Kra identity
- Bijective proof
- Warnaar’s bijection
Mathematics Subject Classification (2010)
- 05A17
- 05A19
- 11P83
- 05A15