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Counting Certain Pairings in Arbitrary Groups

  • Y. O. HAMIDOUNE (a1)

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.

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[1]Chowla S. (1935) A theorem on the addition of residue classes: Applications to the number Γ(k) in Waring's problem. Proc. Indian Acad. Sci. 2 242243.
[2]Eliahou S. and Lecouvey C. (2008) Matchings in arbitrary groups. Adv. Appl. Math. 40 219224.
[3]Erdős P. and Heilbronn H. (1964) On the addition of residue classes mod p. Acta Arith. 9 149159.
[4]Fan C. K. and Losonczy J. (1996) Matchings and canonical forms in symmetric tensors. Adv. Math. 117 228238.
[5]Fournier J. C. (2003) Combinatorics of perfect matchings in plane bipartite graphs and application to tilings. Theoret. Comput. Sci. 303 333351.
[6]Hall P. (1935) On representatives of subsets. J. London Math. Soc. 10 2630.
[7]Hamidoune Y. O. (1984) On the connectivity of Cayley digraphs. Europ. J. Combin. 5 309312.
[8]Hamidoune Y. O. (1996) An isoperimetric method in additive theory. J. Algebra 179 622630.
[9]Hamidoune Y. O. (1997) On subsets with a small sum in abelian groups I: The Vosper property. Europ. J. Combin. 18 541556.
[10]Hamidoune Y. O. (1999) On small subset product in a group. In Structure Theory of Set-Addition, Vol. 258 of Astérisque, pp. 281–308.
[11]Hamidoune Y. O. (2000) Some results in additive number theory I: The critical pair theory. Acta Arith. 96 97119.
[12]Hamidoune Y. O. (2008) On group bijections φ with φ(B) = A and ∀aB, aφ(a) ∉ A. arXiv:0812.2522.
[13]Hamidoune Y. O. Hyper-atoms and the Kemperman's critical pair theory, arXiv.0708.3581.
[14]Hamidoune Y. O. Hyper-atoms applied to the critical pair theory. Submitted. arXiv:1102.2099v1
[15]Kemperman J. H. B. (1956) On complexes in a semigroup. Nederl. Akad. Wetensch. Proc. Ser. A 59 247254.
[16]Kneser M. (1953) Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58 459484.
[17]Losonczy J. (1998) On matchings in groups. Adv. Appl. Math. 20 385391.
[18]Lovász L. and Plummer M. D. (1986) Matching theory. Ann. Discrete Math. 29.
[19]Olson J. E. (1975/76) Sums of sets of group elements. Acta Arith. 28 147156.
[20]Olson J. E. (1984) On the sum of two sets in a group. J. Number Theory 18 110120.
[21]Plagne A. (2011) Yahya ould Hamidoune, grand Mauritanien, homme singulier, mathématicien d'exception. Gaz. Math. 129 123129.
[22]Plagne A. (2011) Yahya ould Hamidoune, the Mauritanian mathematician. Combin. Probab. Comput. 20 641645.
[23]Scherk P. and Moser L. (1955) Advanced problems and solutions: Solutions, 4466. Amer. Math. Monthly 62 4647.
[24]Wakeford E. K. (1918/1919) On canonical forms. Proc. London Math. Soc. 18 403410.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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