Ayrışımların Modulo 11 Kongrüans Özellikleri

Yıl 2011, Cilt: 2 Sayı: 2, 61 - 73, 01.12.2011

Göksal Bilgici

Öz

Atkin and Swinnerton – Dyer calculated some congruences properties of the generating function for the partitions for modulo 5, 7 and 11. They obtained the results directly for modulo 5 and 7, but for modulo 11, they gave the congruences, and proved later by using these congruences. In this paper, it is explained how the congruences given by Atkin and Swinnerton – Dyer can be calculated for modulo 11.

Anahtar Kelimeler

Partitions, Partition Function, Partition Congruences.

Congruences Properties of Partitions for Modulo

Öz

Atkin and Swinnerton – Dyer calculated some congruences properties of the generating function for the partitions for modulo 5, 7 and 11. They obtained the results directly for modulo 5 and 7, but for modulo 11, they gave the congruences, and proved later by using these congruences. In this paper, it is explained how the congruences given by Atkin and Swinnerton – Dyer can be calculated for modulo 11.

Anahtar Kelimeler

Partitions, Partition Function, Partition Congruences.

Kaynakça

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• Dyson, F.J. (1944). Some Guesses in Theory of Partitions. Eureka, Cambridge, 8, 10 – 15,.
• Garvan, F.G. (1988). New Combinatorial Interpretations of Ramanujan's Partition Congruences mod 5,7 and 11. Trans. Amer. Math. Soc., 305, 47—77.
• Hardy, G. H., Ramanujan, S. (1918). Asymptotic Formulae in Combinatory Analysis. Proc. London Math. Soc., 17, 75-115.
• Hirschhorn, M.D. (1987). A Generalization of Winquist’s Identity and a Conjecture of Ramanujan. Journal of Indian Mathematical Society, 51.
• Hirschorn, M. D. (1999). Another Short Proof of Ramanujan's Mod 5 Partition Congruences and More. Amer. Math. Monthly, 106, 580-583.
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• MacMahon, P. A. (1926). The Parity of p(n), the Number of Partitions of n., when n 1000. J. London Math. Soc. 1, 225-226.
• Ramanujan, S. (1927). Some Properties of p(n), the Number of partitions of n. Paper 25 of Collected Papers of S. Ramanujan, Cambridge University Pres, London.
• Skiena, S. (1990). Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley.
• Winquist, L. (1968). An Elementary Proof of p (11m 
• (mod 11). J. Combinatorial 6) Theory, 6, 56-69.