Abstract
In this paper we present a new formula for the number of unrestricted partitions of n. We do this by introducing a correspondence between the number of unrestrited partitions of n and the number of non-negative solutions of systems of two equations, involving natural numbers in the interval \((1 ,n^{2})\).
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References
Andrews, G.E.: Generalized Frobenius partitions. Mem. Am. Math Soc. 49, 1–44 (1984)
Brietzke, E.H.M., Santos, J.P.O., da Silva, R.: A new approach and generalizations to some results about mock theta functions. Discrete Math. 311(8), 595–615 (2011)
Brietzke, E.H.M., Santos, J.P.O., da Silva, R.: Combinatorial interpretations as two-line array for the mock theta functions. Bull. Braz. Math. Soc. New Series 44(2), 233–253 (2013)
Bolker, E.D.: Elementary Number Theory. W.A. Benjamin, Inc., New York (1970)
Dickson, L.E.: Polygonal numbers and related Warings problems. Q. J. Math. 5, 283–290 (1934)
Dickson, L.E.: Two-fold generalizations of Cauchy’s lemma. Am. J. Math. 56, 513–528 (1934)
Frobenius, G.: Uber die Charaktere der Symmetrischen Gruppe, pp. 516–534. Sitzber. Pruess. Akad., Berlin (1900)
Kloosterman, H.D.: Simultane Darstellung zweier ganzen Zahlen als einer Summe von ganzen Zahlen und deren Quadratsumme. Math. Ann. 18, 319–364 (1942)
Matte, M.L., Santos, J.P.O.: A new approach to integer partitions. Bull. Braz. Math. Soc. New Series 49(4), 811–847 (2018)
Mondek, P., Ribeiro, A.C., Santos, J.P.O.: New two-line arrays representing partitions. Ann. Comb. 15, 341–354 (2011)
Landau, L.: Elementary Number Theory. Chelsea Publishing Co, New York (1958)
Pall, G.: Simultaneous quadratic and linear representation. Q. J. Math. 2, 136–143 (1931)
Acknowledgements
The authors would like to express their gratitute to Professors Robson da Silva and Eduardo Brietzke for their valuable comments on an early draft of this paper.
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This paper was written while the first author enjoyed the hospitality of the Universidade de Campinas in São Paulo-Brazil, supported by a grant from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico)-Brazil.
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Godinho, H., Santos, J.P.O. On a new formula for the number of unrestricted partitions. Ramanujan J 55, 297–307 (2021). https://doi.org/10.1007/s11139-019-00211-7
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DOI: https://doi.org/10.1007/s11139-019-00211-7