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Partitions into large unequal parts from a general sequence

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  09 April 2009

Kevin John Fergusson
Kevin John Fergusson
Affiliation:
17 Blackdown way, Karrinyup WA 6018, Australia, e-mail: kevinjohnfergusson@hotmail.com
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Abstract

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An asymptotic estimate is obtained for the number of partitions of the positive integer n into unequal parts coming from a sequence u, with each part greater than m, under suitable conditions on the sequence u. The estimate holds uniformly with respect to integers m such that 0 ≤ m ≤ n1−δ, as n → ∞, where δ is a given real number, such that 0 < δ < 1.

MSC classification

Secondary: 11P82: Analytic theory of partitions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 13 - 44
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Ash, R., Real analysis and probability (Academic Press, New York, 1972).Google Scholar
[2]Cassels, J. W. S., ‘On the representation of integers as the sums of distinct summands taken from a fixed set’, Acta Sci. Math. Szeged 21 (1960), 111–124.Google Scholar
[3]Erdós, P., Nicolas, J. and Szalay, M., Partitions into parts which are unequal and large Lecture Notes in Math. 1380 (Springer, Berlin, 1989) pp. 19–30.Google Scholar
[4]Fergusson, K., ‘Partitions into k–th powers’, preprint.Google Scholar
[5]Fergusson, K., Partitions into unequal large parts (Ph.D. Thesis, The University of Adelaide, Adelaide, South Australia, 1996).Google Scholar
[6]Freiman, G., ‘On extremal additive problems of Paul Erdős’, Ars Combin. 26B (1988), 93–114.Google Scholar
[7]Freiman, G. and Pitman, J., ‘Partitions into distinct large parts’, J. Austral. Math. Soc. 57 (1994), 386–416.Google Scholar
[8]Gnedenko, B. and Kolmogorov, A., Limit distributions for sums of independent random variables (Addison-Wesley, Cambridge, Massachusetts, 1954).Google Scholar
[9]Hardy, G. and Ramanujan, S., ‘Asymptotic formulae in combinatory analysis’, Proc. London Math. Soc. 17 (1918), 75–115.Google Scholar
[10]Hua, L., ‘On the number of partitions of a number into unequal parts’, Trans. Amer. Math. Soc. 51 (1942), 194–201.Google Scholar
[11]Ingham, A. E., The distribution of prime numbers, Cambridge Tracts in Mathematics 30 (Cambridge University Press, London, 1932).Google Scholar
[12]Ingham, A. E., ‘A Tauberian theorem for partitions’, Ann. of Math. (2) 42 (1941), 1075–1090.CrossRefGoogle Scholar
[13]Postnikov, A., Introduction to analytic number theory, Translations of Mathematical monographs 68 (American Mathematical Society, Providence, RI, 1988).CrossRefGoogle Scholar
[14]Roth, K. and Szekeres, G., ‘Some asymptotic formulae in the theory of partitions’, Quart. J. Math. (Oxford) (2) 5 (1954), 241–259.CrossRefGoogle Scholar
[15]Spivak, M., Calculus, 2nd edition (Publish or Perish, Berkeley, CA, 1980).Google Scholar
[16]Uspensky, J., ‘Asymptotic expressions of numerical functions occurring in problems of partition of numbers’, Bull. Acad. Sci. de Russie 14 (1920), 199–218.Google Scholar