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Pré-Publication, Document De Travail Année : 2015

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 $n \ge 4$.
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hal-01120478 , version 1 (25-02-2015)

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Eric Balandraud, Maurice Queyranne, Fabio Tardella. LARGEST MINIMAL INVERSION-COMPLETE AND PAIR-COMPLETE SETS OF PERMUTATIONS. 2015. ⟨hal-01120478⟩
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