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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 33, Pages 1725-1735
http://dx.doi.org/10.1155/S0161171204307258

Product partitions and recursion formulae

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received 30 July 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Utilizing a method briefly hinted in the author's paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functions p*(n;k,h) (which stands for the number of product partitions of n>1 into k parts of which h are distinct), and p(d)*(n;m) (which stands for the number of product partitions of n into exactly m parts with at most d repetitions of any part). We also derive recursion formulae for certain product partition functions without the use of generating functions.