Largest minimal inversion-complete and pair-complete sets of permutations

TL;DR

This paper determines the maximum size of minimal permutation sets that cover all inversions or pairs, revealing that these sizes are equal for all n ≥ 4, using graph theory and combinatorial analysis.

Contribution

It solves the extremal problem of finding maximum sizes of minimal inversion- and pair-complete permutation sets, providing explicit values and descriptions.

Findings

Maximum cardinalities are determined using Mantel's Theorem.

The maximum sizes for inversion- and pair-complete sets coincide for n ≥ 4.

Complete descriptions of optimal sets are provided.

Abstract

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 $1 \le i \textless{} j \le n$, (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 $n \ge 4$.