Find the number of ways of seating n people A1 , A2 , ..., An in arbitrary number of circular tables of arbitrary sizes.
Yes, Circular permutations do not matter. For example, Around a circle A,B,C are three people who sit as ABC, BCA, CAB clockwise are considered equivalent and counted as just one configuration.
And tables are indistinguishable. So, only the configuration of people around the tables matters.
Here is how I counted for n = 4:
4 can be written as,
4 = (4-1)! ways = 6 (4 people around 1 table)
3+1 = (3-1)!*(1-1)! ways = 2 (4 people around 2 tables etc)
2+2 = (2-1)!*(2-1)! ways = 1
2+1+1 = (2-1)!(1-1)!(1-1)! ways = 1
1+1+1+1 = (1-1)!(1-1)!(1-1)!*(1-1)! ways = 1
so total I could count only 11 ways.(Maybe this is flawed in some way? because others think the number of configurations is n! )