An elementary solution to a problem in restricted partitions

Abstract

The present paper deals with an apparently hitherto untreated problem in the theory of restricted partitions: What is the number T_n^(r) of distinct partitions of the composite integer nr that can be made by partwise addition of n—not necessarily distinct—partitions of r? The answer is given in the form of a finite series of binomial coefficients multiplied by certain integer coefficients which dependend only on r:

$T_n^{\left(r\right)}=\sum _{i=0}^{r-g-1}{c}_i^{\left(r\right)}\left(\begin{array}{c}n+g\\ g+i\end{array}\right),whereg=\left[\frac{r+1}{2}\right]$

In general the non-vanishing c_i^(r) must be determined by direct calculation; in this paper we give them for all r≤11. Several other interpretations of T_n^(r) are given, and some additional open questions concerning the interpretation of the results are discussed.