I must tell you that I have no idea what you post is saying.
However Stirling numbers of the second kind and Bell Numbers may be of some use to you.
I would like to know how to obtain the cardinality of a set of partitions for the given cardinality of another set. For example, given a set of 4 elements, I obtain the following set of 5 partitions with their subset cardinalities ranging over indices from 1 to 4:0 0 0 1I'd like to know the name of this set too. Thank you in advance for your help.
0 2 0 0
1 0 1 0
2 1 0 0
4 0 0 0