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The arithmetic of partitions into distinct parts

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Scott Ahlgren  and
Jeremy Lovejoy
Scott Ahlgren
Affiliation:
Department of Mathematics, Colgate University, Hamilton, NY 13346, U.S.A. Current address: Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A. E-mail: [email protected]
Jeremy Lovejoy
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A. E-mail: [email protected]
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Extract

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function

MSC classification

Secondary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 203 - 211
Copyright
Copyright © University College London 2001

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References

1.Ahlgren, S.. The distribution of the partition function modulo composite integers M. Math. Ann., 318 (2000), 795–803.CrossRefGoogle Scholar
2.Ahlgren, S. and Ono, K.. Congruence properties for the partition function, submitted.Google Scholar
3.Gordon, B. and Hughes, K.. Multiplicative properties of η-products II. Cont. Math., 143 (1993), 415–430.Google Scholar
4.Gordon, B. and Hughes, K.. Ramanujan congruences for q(n). Analytic Number Theory, Lecture Notes in Math., 899, Springer (New York, 1981), 333–359.CrossRefGoogle Scholar
5.Gordon, B. and Ono, K.. Divisibility of certain partition functions by powers of primes. Ramanujan J. 1 (1997), 25–34.CrossRefGoogle Scholar
6.Koblitz, N.. Introduction to Elliptic Curves and Modular Forms. Springer-Verlag (New York, 1984).CrossRefGoogle Scholar
7.Lovejoy, J.. The divisibility and distribution of partitions into distinct parts. Advances Math., 158(2001), 253–263.CrossRefGoogle Scholar
8.Ono, K.. Distribution of the partition function modulo m. Ann. Math. 151 (2000), 1–15.Google Scholar
9.Ono, K. and Penniston, D.. The 2-adic behavior of the number of partitions into distinct parts. J. Combinat. Theory Ser. A, 92 (2000), 138–157.CrossRefGoogle Scholar
10.Rickert, J.. Divisibility of restricted partition functions (preprint).Google Scholar
11.Rødseth, Øystein. Congruence properties of the partition functions q(n) and q 0(n). Arbok Univ. Bergen Mat.—Natur. Ser. (1969), no. 13.Google Scholar
12.Serre, J. P.. Divisibilité de certaines fonctions arithmétiques. L'Ensign. Math., 22 (1976), 227–260.Google Scholar
13.Sturm, J., On the congruence of modular forms. Springer Lect. Notes in Math., 1240 (1984), 275–280.Google Scholar