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Counting Certain Pairings in Arbitrary Groups

Published online by Cambridge University Press:  11 October 2011

Y. O. HAMIDOUNE
Y. O. HAMIDOUNE*
Affiliation:
Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu, Combinatoire et Optimisation, Case 189, 4 Place Jussieu, 75005 Paris, France
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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 bφ(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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 855 - 865
Copyright
Copyright © Cambridge University Press 2011

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