Matrix representation of a partition

Question

Is there a natural way to represent all the partitions of an integer set $\{1,2,3,...,n\}$ as a matrix in the similar way permutations can be mapped to group of matrices?

Answer 1

The important component of the question here seems to be: can we define a natural binary operation on the set P of partitions of $\\{1,2,\ldots,n\\}$ to form a group G? Cayley's Theorem tells us that if G exists, it is isomorphic to a subgroup of the symmetric group on G. So G will be isomorphic to a group of permutation matrices.

However, finding a natural binary operation on P is going to be tricky. Of course, P is just a set, and it would be possible to construct some highly-contrived binary operation on P, but typically it would not preserve the structure of the partitions.

As an off-the-top-of-my-head example of why I think it should be tricky: |P| is given by the Bell Numbers, which can be a prime number, whence G must be a cyclic group and each non-identity element of G must somehow generate G.