On the number of partitions into odd parts or congruent to $\pm 2 \pmod{10}$

Mircea Merca

Abstract


Let $R_2(n)$ denote the number of partitions of $n$ into parts that are odd or congruent to $\pm 2 \pmod{10}$. In 2007, Andrews considered partitions with some negative parts and provided a second combinatorial interpretation for $R_2(n)$. In this paper, we give a collection of linear recurrence relations for the partition function $R_2(n)$. As a corollary, we obtain a simple criterion for deciding whether $R_2(n)$ is odd or even. Some identities involving overpartitions and partitions into distinct parts are derived in this context.

Keywords


integer partitions, partition congruences, recurrence relations

Full Text:

PDF

References


G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Co., 1976.

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc. (N.S.) 44 (2007), 561-573.

G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), 1523-1533.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge University Press, Cambridge, 2004.

L. Carlitz and M. V. Subbarao, A simple proof of the quintuple product identity, Proc. Amer. Math. Soc. 32 (1972), 42-44.

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

G. Gaspar and M. Rahmana, Basic Hypergeometric Series, Cambridge University Press, 1990.

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

B. Kim, A short note on the overpartition function, Discrete Math. 309 (2009), 2528-2532.

J. Lovejoy, Gordon's theorem for overpartitions, J. Comb. Theory Ser. A 103 (2003), 393-401.

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

J. Lovejoy, Overpartitions and real quadratic Fields, J. Number Theory 106 (2004), 178-186.

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

M. Merca, New relations for the number of partitions with distinct even parts, J. Number Theory 176 (2017), 1-12.

N. J. A. Sloane, The online encyclopedia of integer sequences., Published electronically at http://oeis.org, 2016.

M. V. Subbarao and M. Vidyasagar, On Watson's quintuple product identity, Proc. Amer. Math. Soc. 26 (1970), 23-27.




Contributions to Discrete Mathematics. ISSN: 1715-0868