# Consider a set of 64 elements with subsets that have cardinalities 1, 1, 2, 5, 5, 10,10,10,20

Consider a set of 64 elements exhausted by subsets that have cardinalities 1, 1, 2, 5, 5, 10,10,10,20.

Questions:

1) How many different ways are there to gather these subsets into two new subsets with 32 elements each?

2) Is this a specific example of a "standard" general problem in combinatorics/combinations that is solved by a "standard" general technique?

3) If so, can you describe or provide a link to the technique?

4) In this case the subset cardinalities have an obvious triplet structure:

1+1 = 2

5+5 = 10

10+10 = 20

Is this particular triplet subset cardinality structure an example of any more general kind of triplet subset cardinality structure that has been previously studied?

Thanks as always for whatever time you can afford to spend considering this matter.

## 1 Answer

The normal approach would be by generating functions. You would want the coefficient of $x^{32}$ in $(1+x)^2(1+x^2)(1+x^5)^2(1+x^{10})^3(1+x^{20})$ where each factor represents taking or not taking one subset. I am not aware of any study of your triplet structure.

• thank you so much time for taking the time to respond. I will pass your answer on to my team's senior statistical member (he may well be aware of it already - I haven't asked him yet because he's busy on another project-related task.) One other question - if I could show a convincing empirical example of where this triplet structure occurs in nature, do you think anyone would be interested in generating a formal treatment of the matter? – David Halitsky Dec 9 '17 at 2:21
• TobiasKildetoft - the reason I asked this question is because the specific cardinalities 1,1,2,5,5,10,10,20 are intimately related to the way in which we derive the root system of $E_8$. Therefore, I was very happy to see that 1,2,5,10,20 is well-known with a number of important references in OEIS: oeis.org/… – David Halitsky Dec 9 '17 at 3:45
• RossMillikan - see my comment above to Tobias . . . – David Halitsky Dec 9 '17 at 3:46