Bounded ranges of cardinal functions

Let x be a (non-empty) sequence of positive real numbers. Its achievement set x is the set of all the possible sums of the elements of x. The cardinal function of x is the function f:A(x) → N∪\ω,c\ that for every x(x) the value $f(x)$ is equal to the number of ways x is represented as a sum of elements of x. In this paper we consider possible ranges of cardinal functions of sequences x. We present some general constructions and several criteria that a set has to satisfy in order to be a range of a cardinal function. We put special attention to the case of sets with maximal element equal to $6$. In this case, in particular, we obtained a full characterisation of sets that are ranges of cardinal functions of interval-filling sequences.