LARGEST MINIMAL INVERSION-COMPLETE AND PAIR-COMPLETE SETS OF PERMUTATIONS
Résumé
We solve two related extremal problems in the theory of permutations.
A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete)
if for every inversion $(j,i)$, where $1 \le i < j \le n$, (resp., for every pair $(i,j)$, where $i\not= j$)
there exists a permutation in~$Q$ where $j$ is before~$i$.
It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness.
The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and
that of a minimal pair-complete set of permutations.
The latter problem arises in the determination of the Carath\'eodory numbers for
certain abstract convexity structures on the $(n-1)$-dimensional real and integer vector spaces.
Using Mantel's Theorem on the maximum number of edges in a triangle-free graph,
we determine these two maximum cardinalities and
we present a complete description of the optimal sets of permutations for each problem.
Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever $n \ge 4$.
Domaines
Combinatoire [math.CO]
Origine : Fichiers produits par l'(les) auteur(s)