A Note on Overcubic Partitions

From NZJM

Jump to: navigation, search

New Zealand Journal of Mathematics

Vol. 42, (2012), Pages 229-234


Michael D. Hirschhorn

Department of Mathematics and Statistics

UNSW

Sydney

Australia 2052

mailto:m.hirschhorn@unsw.edu.au






Abstract Byungchan Kim introduced the notion of overcubic partitions of the number n, partitions of n in which odd parts come in two colours, one of which can occur at most once, and in which the even parts come in four colours, two of which can occur at most once each, and let the number of overcubic partitions of $n$ be \overline{a}(n). Thus, for example, \overline{a}(4)=26, since

4 = 41 = 42 = 43 = 44 = 31 + 11 = 31 + 12 = 32 + 11 = 32 + 12 = 21 + 21 = 21 + 22

= 21 + 23 = 21 + 24 = 22 + 22 = 22 + 23 = 22 + 24 = 23 + 24 = 21 + 11 + 11

= 21 + 11 + 12 = 22 + 11 + 11 = 22 + 11 + 12 = 23 + 11 + 11 = 23 + 11 + 12

= 24 + 11 + 11 = 24 + 11 + 12 = 11 + 11 + 11 + 11 = 11 + 11 + 11 + 12.

Using the theory of modular functions, Kim showed that \sum_{n\ge0}\overline{a}(3n+2)q^n=6\,\prod_{n\ge1}\frac{(1-q^{3n})^6(1-q^{4n})^3}{(1-q^n)^8(1-q^{2n})^3}. Of course, this implies \overline{a}(3n+2)\equiv0\pmod6. Using nothing more than Jacobi's triple product identity, we prove Kim's identity and obtain expressions for \sum_{n\ge0}\overline{a}(3n)q^n and \sum_{n\ge0}\overline{a}(3n+1)q^n (which are not simple products) as sums of two products.

Keywords Overcubic partitions; 3-dissection.

Classification (MSC2000) 11P81, 11P83.

Full text

Full paper

Personal tools