Permutation Capacities of Families of Oriented Infinite Paths

  • Authors:

  • Graham Brightwell;Gérard Cohen;Emanuela Fachini;Marianne Fairthorne;János Körner;Gábor Simonyi;Ágnes Tóth

  • Affiliations:

  • G.R.Brightwell@lse.ac.uk and M.I.Fairthorne@lse.ac.uk;cohen@enst.fr;fachini@di.uniroma1.it and korner@di.uniroma1.it;-;-;simonyi@renyi.hu;tothagi@cs.bme.hu

  • Venue:
  • SIAM Journal on Discrete Mathematics
  • Year:
  • 2010

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Abstract

Körner and Malvenuto asked whether one can find $\binom{n}{\lfloor n/2\rfloor}$ linear orderings (i.e., permutations) of the first $n$ natural numbers such that any pair of them places two consecutive integers somewhere in the same position. This led to the notion of graph-different permutations. We extend this concept to directed graphs, focusing on orientations of the semi-infinite path whose edges connect consecutive natural numbers. Our main result shows that the maximum number of permutations satisfying all the pairwise conditions associated with all of the various orientations of this path is exponentially smaller, for any single orientation, than the maximum number of those permutations which satisfy the corresponding pairwise relationship. This is in sharp contrast to a result of Gargano, Körner, and Vaccaro concerning the analogous notion of Sperner capacity of families of finite graphs. We improve the exponential lower bound for the original problem and list a number of open questions.