Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

Permutations Destroying Arithmetic Structure

Given a linear form C1X1 + · · · + CnXn, with coefficients in the integers, we characterize exactly the countably infinite abelian groups G for which there exists a permutation f that maps all solutions (α1, . . . , αn) ∈ Gn (with the αi not all equal) to the equation C1X1+ · · ·+CnXn = 0 to non-solutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arit...

Permutations that Destroy Arithmetic Progressions in Elementary p-Groups

Given an abelian group G, it is natural to ask whether there exists a permutation π of G that “destroys” all nontrivial 3-term arithmetic progressions (APs), in the sense that π(b)− π(a) 6= π(c)− π(b) for every ordered triple (a, b, c) ∈ G3 satisfying b − a = c − b 6= 0. This question was resolved for infinite groups G by Hegarty, who showed that there exists an AP-destroying permutation of G i...

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of 1 2 on the lower density of subsets of positive inte...

Enumerating Permutations that Avoid Three Term Arithmetic Progressions

It is proved that the number of permutations of the set {1, 2, 3, . . . , n} that avoid three term arithmetic progressions is at most (2.7) n 21 for n ≥ 11 and at each end of any such permutation, at least ⌊ 2 ⌋−6 entries have the same parity.

Permutations Avoiding Arithmetic Progressions