The Abstract

	Annals of Combinatorics 10 (2006) Arithmetic Properties of Overpartitions into Odd Parts Michael D. Hirschhorn and James A. Sellers School of Mathematics, University of New South Wales, Sydney 2052, Australia m.hirschhorn@unsw.edu.au Department of Mathematics, the Pennsylvania State University, University Park, PA 16802, USA sellersj@math.psu.edu Received March 17, 2005 AMS Subject Classification: 05A17, 11P83 Abstract. In this article, we consider various arithmetic properties of the function ‾p₀(n) 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 ‾p₀(n) and some easily-stated characterizations of ‾p₀(n) modulo small powers of two. For example, it is proven that, for n ≥ 1, ‾p₀(n) ≡ 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) 617–636. 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, Trends Math. (2004) 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. Get the DVI \| PS \| PDF file of this abstract. back

Annals of Combinatorics 10 (2006)

Arithmetic Properties of Overpartitions into Odd Parts

Michael D. Hirschhorn and James A. Sellers

School of Mathematics, University of New South Wales, Sydney 2052, Australia
m.hirschhorn@unsw.edu.au

Department of Mathematics, the Pennsylvania State University, University Park, PA 16802, USA
sellersj@math.psu.edu

Received March 17, 2005

AMS Subject Classification: 05A17, 11P83

Abstract. In this article, we consider various arithmetic properties of the function ‾p₀(n) 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 ‾p₀(n) and some easily-stated characterizations of ‾p₀(n) modulo small powers of two. For example, it is proven that, for n ≥ 1, ‾p₀(n) ≡ 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) 617–636.

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, Trends Math. (2004) 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.

Get the DVI | PS | PDF file of this abstract.

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