Skip to main content
×
Home
    • Aa
    • Aa

Counting Certain Pairings in Arbitrary Groups

  • Y. O. HAMIDOUNE (a1)
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 : BA such that bφ(b) ∉ A, for every bB. 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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] S. Eliahou and C. Lecouvey (2008) Matchings in arbitrary groups. Adv. Appl. Math. 40 219224.

[4] C. K. Fan and J. Losonczy (1996) Matchings and canonical forms in symmetric tensors. Adv. Math. 117 228238.

[5] J. C. Fournier (2003) Combinatorics of perfect matchings in plane bipartite graphs and application to tilings. Theoret. Comput. Sci. 303 333351.

[7] Y. O. Hamidoune (1984) On the connectivity of Cayley digraphs. Europ. J. Combin. 5 309312.

[8] Y. O. Hamidoune (1996) An isoperimetric method in additive theory. J. Algebra 179 622630.

[9] Y. O. Hamidoune (1997) On subsets with a small sum in abelian groups I: The Vosper property. Europ. J. Combin. 18 541556.

[11] Y. O. Hamidoune (2000) Some results in additive number theory I: The critical pair theory. Acta Arith. 96 97119.

[15] J. H. B. Kemperman (1956) On complexes in a semigroup. Nederl. Akad. Wetensch. Proc. Ser. A 59 247254.

[16] M. Kneser (1953) Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58 459484.

[17] J. Losonczy (1998) On matchings in groups. Adv. Appl. Math. 20 385391.

[20] J. E. Olson (1984) On the sum of two sets in a group. J. Number Theory 18 110120.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 39 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 30th May 2017. This data will be updated every 24 hours.