Definition. A subset of is said to be -dense if such that .
Lemma. The number of compositions of with not more than parts and no part exceeding is defined as . By Inclusion-Exclusion principle,
Definition. is defined as the number of -dense subsets of with least cardinality.
We observe that this least cardinality has to be .
2. The open problem
The following matrix shows the values with .
Can we arrive at the values and the values of ‘s where the function maximizes?
(I will write more soon 🙂 )