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?

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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?

Leslie Rollins · Accepted Answer

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    ,    …    ,    n       to form a group   G?  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.

Is there a natural way to represent all the partitions of an integer set <mo fence="false" stret

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