P -orderings of finite subsets of Dedekind domains

If R is a Dedekind domain, P a prime ideal of R and S ⊆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 {ai}i=1 of the elements of S with the property that, for each 1 < i ≤m, the choice of ai minimizes the P -adic valuation of ∏j<i(s− aj ) over elements s ∈ S. If S, S′ are two finite subsets of R of the same cardinality then a bijection φ : 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.