P-orderings of finite subsets of Dedekind domains

If R is a Dedekind domain, P a prime ideal of R and S subset of R a finite subset then a P-ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490: 101-127, 1997), is an ordering {a(i)}(i=1)(m) of the elements of S with the property that, for each 1 < i <= m, the choice of a(i) minimizes the P-adic valuation of Pi(j<i) (s - a(j)) over elements s is an element of S. If S, S' are two finite subsets of R of the same cardinality then a bijection phi : S -> S' is a P-ordering equivalence if it preserves P-orderings. In this paper we give upper and lower bounds for the number of distinct P-orderings a finite set can have in terms of its cardinality and give an upper bound on the number of P-ordering equivalence classes of a given cardinality.