# Is it possible to quantify the number of dimensions in

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.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Blaine Foster
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. There's no point in having a notion of "dimension" unless you can use it for something.