Counting Certain Pairings in Arbitrary Groups

Source abstract

In this paper, we study certain pairings which are defined as follows: if A and B are finite subsets of an arbitrary group, a Wakeford–Fan–Losonczy pairing from B onto A is a bijection φ : B → A such that b φ( b ) ∉ A , for every b ∈ B . The number of such pairings is denoted by μ( B, A ). We investigate the quantity μ( B, A ) for A and B , two finite subsets of an arbitrary group satisfying 1 ∉ B , | A | = | B |, and the fact that the order of every element of B is ≥ | B | + 1. Extending earlier results, we show that in this case, μ( B, A ) is never equal to 0. Furthermore we prove an explicit lower bound on μ( B, A ) in terms of | B | and the cardinality of the group generated by B , which is valid unless A and B have a special form explicitly described. In the case A = B , our bound holds unless B is a translate of a progression.