Efficient reconstruction of partitions

We consider the problem of reconstructing a partition x of the integer n from the set of its t-subpartitions. These are the partitions of the integer n-t obtained by deleting a total of t from the parts of x in all possible ways. It was shown (in a forthcoming paper) that all partitions of n can be reconstructed from t-subpartitions if n is sufficiently large in relation to t. In this paper we deal with efficient reconstruction, in the following sense: if all partitions of n are t--reconstructible, what is the minimum number N=N-(n,t) such that every partition of n can be identified from any N+1 distinct subpartitions? We determine the function N-(n,t) and describe the corresponding algorithm for reconstruction. Superpartitions may be defined in a similar fashion and we determine also the maximum number N+(n,t) of t-superpartitions common to two distinct partitions of n.