<Emphasis Type="Italic">P</Emphasis>-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 {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 ∏ _j<i (s−a _j ) 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.