Abstract.
For 
k1, let 
p k (n) count the number of 
k-component multipartitions of a nonnegative integer 
n, and let 
(n)= dn d be the usual divisor function. In this paper, we give a combinatorial proof of the recursive formula
p k (n)=k n r=1 n p k (n-r)(r),
k1, where 
p k (n) is defined as above, and also for 
k<0, which requires a subtler approach.
This formula was used by Gandhi in 1963 to prove several theorems, which yield numerous Ramanujan type congruences for 
p k (n), including some well-known congruences for Ramanujan's 
-function.
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