## Abstract

The first problem solved here is that of determining the coefficient of any S-function {$\lambda $} in the expansion of {m} $\otimes $ {$\mu $}, where m is a positive integer and ($\mu $) is a partition of 4. The method is not recursive or laborious and can be applied equally well to large as to small values of m. It will also yield a specific formula for the coefficient when {$\lambda $} has any prescribed form. Such formulae given here include those for the coefficients of {4m-k, k}, {m+k, m+k, m-k, m-k}, {m+k, m, m, m-k}, {4m-2k, k, k}, where m $\geq $ k $\geq $ 0, and several other types. Some of the results proved incidentally in the development of this method are also of some intrinsic interest. Thus a formula is obtained and proved for the coefficient of {$\lambda $} = {$\lambda _{1}$, $\lambda _{2}$,$\ldots $, $\lambda _{n}$} in {m}$^{n}$ when $\lambda _{2}\leq $ m and n is any integer, and it is proved that the coefficient of any {$\lambda $} in {m}$^{n}$ is congruent to 1, 0 or -1, mod n when n is prime and is congruent to 1, 0 or -1, mod n -1 when n -1 is prime. From the results obtained for ($\mu $) a partition of four certain S-functions are seen to have the same coefficient in {m} $\otimes $ {$\mu $}. Thus, if $\lambda _{1}\leq $ 2m, then the coefficients of {$\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$, $\lambda _{4}$} and {2m-$\lambda _{4}$, 2m-$\lambda _{3}$, 2m-$\lambda _{2}$, 2m-$\lambda _{1}$} are proved to be equal, and if $\beta $ is even, $\lambda _{2}\leq $ m, and m+$\lambda _{4}\geq \beta \geq \lambda _{2}$ then the coefficients of {$\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$, $\lambda _{4}$} and {8m-3$\beta $-$\lambda _{1}$, $\beta $-$\lambda _{4}$, $\beta $-$\lambda _{3}$, $\beta $-$\lambda _{2}$} are equal. These results on related coefficients are gathered into five main theorems which are proved for all m and n in the last section of the paper.