On the existence of p̄-core partitions of natural numbers

Abstract. Extending previous work of J. B. Olsson, cf. [7], [8], and of K. Erdmann and G. O. Michler, cf. [2], on the number of p-spin blocks of defect zero (p prime) of a double covering group of the symmetric group Sn, we prove that this number is positive for all n whenever p ≥ 7. More precisely, it is shown that sp(n)> 0 if p ≥ 7, where sp(n) denotes the number of bar partitions of n which are p̄-cores. 1. Introduction. Everywhere in this article the following notation is used: p denotes an odd prime number, n a natural number, and we put: t: = (p − 1)/2. If G is a finite group, there is some interest in the question whether G has a p-block of defect zero, since the existence of such a block means the existence o