Quantifying dimensionality of combinatorial space Is it possible to quantify the number of dimensions in combinatorial spaces.  The space I am particularly interested is all partitions of a set, bounded by the Bell number, where objects in this space are particular partitions.
 A: It makes sense to consider some sets of combinatorial objects as spaces (or polytopes) and therefore discuss dimensionality (e.g. the set of n by n (-1,0,+1)-matrices).  Although, perhaps the word "dimension" could better be described as "degrees of freedom".

Mathematically, degrees of freedom is
  the dimension of the domain of a
  random vector, or essentially the
  number of 'free' components: how many
  components need to be known before the
  vector is fully determined.

I suspect that it will be difficult to discuss dimensionality in many combinatorial settings.  For example, imagine constructing a Latin square, starting from an empty matrix, placing symbols one-at-a-time in a non-clashing manner.  After placing (say) half of the symbols, we might find: (a) there are still many completions of this partial Latin square, (b) there are no completions of this partial Latin square or (c) there is a unique completion of this partial Latin square.  This seems to go against the notion of dimensionality -- the number of "components" required to determine the Latin square is not fixed.
You could think of the set of partitions of a set of n elements as having dimension n.  You require n pieces of information to determine the partition (which set each element is in).  But as Qiaochu Yuan points out, who cares?  There's no point in having a notion of "dimension" unless you can use it for something.
