Permutations Avoiding Arithmetic Patterns

A permutation π of an abelian group G (that is, a bijection from G to itself) will be said to avoid arithmetic progressions if there does not exist any triple (a, b, c) of elements of G, not all equal, such that c − b = b − a and π(c) − π(b) = π(b) − π(a). The basic question is, which abelian groups possess such a permutation? This and problems of a similar nature will be considered. 1 Notation and Introduction Given a positive integer n, the set {1, ..., n} will be denoted by [n]. If G is a group then Ωn(G) denotes the subset (subgroup when G is abelian) consisting of all elements of G of order dividing n. The cyclic group of order n will be denoted Zn. The symmetric group on n letters will be denoted Sn. In additive and combinatorial number theory, one encounters problems like the following : Let n, k be positive integers with k ≥ 3. How large can a subset A of [n] be which does not contain any k numbers in arithmetic progression? This well-known problem (and a closely related formulation where the set [n] is replaced by the set of all natural numbers N) has a long and distinguished history. For references covering the period up to 1995, see [GGL], Chapter 20. To these should be added the more recent work of Gowers [G1], [G2]. Another much-studied condition is to demand that the set A is a so-called Sidon set, that is, for every quadruple (a, b, c, d) of elements of A, whenever a+ b = c+d then either a = c, b = d or a = d, b = c. Notice that a Sidon set avoids 3-term arithmetic progressions. Note also that the rather similar notion of a so-called (perfect) di erence set has the electronic journal of combinatorics 11 (2004), #R39 1 been extensively studied in general nite abelian (semi)groups, as these have applications to the construction of combinatorial designs (see [CD]). More ambitiously still, one might demand that A has distinct subset sums, that is, given any two distinct subsets of A, the numbers in these two sets have di erent sums. Here there is the well-known conjecture of Erd®s that such a subset of [n] cannot have size greater than log2 n + C, where C is an absolute constant. Erd®s, among others, has also looked at multiplicative analogues of the above problems. Again, we refer the reader to [GGL], Chapter 20, for extensive discussion and references. Speaking informally, one is looking here at sets of integers (or elements in an abelian (semi)group) which avoid a certain kind of ‘arithmetic pattern'. Now in the eld of enumerative combinatorics, the notion of ‘pattern avoidance' has a much more precise meaning. Recall the basic idea Let n ≥ k > 0 be integers, π ∈ Sn, σ ∈ Sk. π is said to avoid σ if there does not exist any k-tuple (a1, ..., ak) of integers such that 1 ≤ a1 < a2 < · · · < ak ≤ n, π(ai) < π(aj) ⇔ σ(i) < σ(j). Here, then, one is looking at permutations of sets of integers which avoid a certain ‘pattern', in the sense just de ned. The basic problem is to count the number of such permutations, as a function of n, for a xed pattern σ. The seminal paper in this more recent area of research is probably [SS]. In this paper we want to combine these two notions of ‘pattern avoidance' to pose new problems. Since this obviously sounds pretty vague, in the next section we will state some precise questions which, after a brief review of the rather sparse relevant literature, will then be the object of study of the rest of the paper. We hope that the discussion above will serve as su cient motivation, and that the reader will also be inspired to think of a multitude of similar questions which one might pose! In this spirit, the paper will include a number of conjectures and open questions which we were unable to resolve, and will be rounded o with some general suggestions for future investigations. 2 Statements of results Henceforth, the words ‘arithmetic progression' will be abbreviated to AP. We begin with a de nition : Definition 2.1 : Let k ≥ 3 be an integer, (S, +) an abelian semigroup and T a subset of S. A permutation π : T → T is said to avoid k-term APs if there does not exist any the electronic journal of combinatorics 11 (2004), #R39 2 k-tuple (a1, ..., ak) of elements of T , not all equal, such that, for i = 1, ..., k − 2, ai + ai+2 = ai+1 + ai+1, and π(ai) + π(ai+2) = π(ai+1) + π(ai+1). On the other hand if such a k-tuple exists, it will be called an AP of length k for π. If k = 3 and the permutation π satis es the requirement of De nition 2.1, we simply say that π avoids APs. The basic question we will be concerned with in this paper is : given S, T and k, does there exist a permutation of T avoiding k-term APs? When the set T is nite, one would naturally also like to ‘count' the number of such permutations, but this issue will not be taken up here.