On m-Ary Overpartitions

Østein J. Røseth^{1} and James A. Sellers^{2}

rodseth@mi.uib.no

sellersj@math.psu.edu

Annals of Combinatorics 9 (3) p.345-353 September, 2005

Abstract:

Presently there is a lot of activity in the study of overpartitions, objects that were
discussed by MacMahon, and which have recently proven useful in several combinatorial studies
of basic hypergeometric series. In this paper we study some similar objects, which we name
m-ary overpartitions. We consider divisibility properties of the number of m-ary overpartitions
of a natural number, and we prove a theorem which is a lifting to general m of the well-known
Churchhouse congruences for the binary partition function.

References:

1. R.F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 (1969) 371--376.

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

3. Ø. J. Røseth and J.A. Sellers, On m-ary partition function congruences: A fresh look at a past problem, J. Number Theory 87 (2001) 270--281.

4. Ø. J. Røseth and J.A. Sellers, Binary partitions revisited, J. Combin. Theory Ser. A 98 (2002) 33--45.