Computing counts of combinations (?)
These combinations of items are sometimes called itemsets.  There are algorithms for finding itemsets that are especially common in a database; see association rule learning.

In general there is no efficient algorithm for producing such a table, as the size of the table can be exponentially large in the size of the input.  Consequently any algorithm will take exponential time on at least some inputs (i.e., will be extremely slow on at least some inputs).  You might want to consider what you plan to do with the table and whether there's a way to achieve your ultimate goal, without building the table explicitly; that might turn out to be a better approach.