Partitions of n that avoid partitions of f, and an application to the tiny-pan coin weighing problem

This paper studies a special form of integer partition of n, where the target is to find a maximal partition P, with the maximum number of parts, that does not include any partition of f as its subset. Let ρ(f,n) denote the number of parts in P. We show that such a problem is closely related to the tiny-pan coin weighing problem where each pan of the balance scale can hold exactly one coin, thereby deriving exact bounds for ρ(f,n) in some of the cases. Furthermore, we show that ρ(f,n) is ultimately periodic as a function of n, so that the computation of ρ(f,n), and its corresponding maximal partition, can be sped up.