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Arithmetic Properties of Overpartitions into Odd Parts
Michael D. Hirschhorn1 and James A. Sellers2
1School of Mathematics, University of New South Wales, Sydney 2052, Australia
m.hirschhorn@unsw.edu.au
2Department of Mathematics, the Pennsylvania State University, University Park, PA 16802, USA
sellersj@math.psu.edu
Annals of Combinatorics 10 (3) p. 353-367 September, 2006
AMS Subject Classification: 05A17, 11P83
Abstract:
In this article, we consider various arithmetic properties of the function which denotes the number of overpartitions of n using only odd parts. This function has arisen in a number of recent papers, but in contexts which are very different from overpartitions. We prove a number of arithmetic results including several Ramanujan-like congruences satisfied by and some easily-stated characterizations of modulo small powers of two. For example, it is proven that, for n ≥ 1, ≡ 0 (mod 4) if and only if n is neither a square nor twice a square.
Keywords: congruence, overpartition, odd parts

References:

1. G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Mass.-London-Amsterdam, 1976.

2. E. Ardonne, R. Kedem, and M. Stone, Filling the Bose sea: symmetric quantum Hall edge states and affine characters, J. Phys. A 38 (2005) 617D636.

3. C. Bessenrodt, On pairs of partitions with steadily decreasing parts, J. Combin. Theory Ser. A 99 (2002) 162–174.

4. S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004) 1623–1635.

5. S. Corteel, J. Lovejoy, and A. Yee, Overpartitions and generating functions for generalized Frobenius partitions, In: Trends in Mathematics, Mathematics and Computer Science III: Algo C rithms, Trees, Combinatorics, and Probabilities, Birkhauser, (2004) pp. 15–24.

6. M.D. Hirschhorn, Partial fractions and four classical theorems of number theory, Amer. Math. Monthly 107 (2000) 260–264.

7. M.D. Hirschhorn, F. Garvan, and J. Borwein, Cubic analogues of the Jacobian theta function q(z, q), Canad. J. Math. 45 (1993) 673–694.

8. M.D. Hirschhorn and J.A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput. 53 (2005) 65–73.

9. J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory Ser. A 103 (2) (2003) 393–401.

10. J. Lovejoy, Overpartition theorems of the Rogers-Ramaujan type, J. London Math. Soc. 69 (2004) 562–574.

11. J. Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004) 178–186.

12. K. Mahlburg, The overpartition function modulo small powers of 2, Discrete Math. 286 (3) (2004) 263–267.

13. J.P.O. Santos and D. Sills, q-Pell sequences and two identities of V. A. Lebesgue, Discrete Math. 257 (1) (2002) 125–142.