On zero-sum free sequences contained in random subsets of finite cyclic groups

Abstract

Let Cn be a cyclic group of order n. A sequence S of length ℓ over Cn is a sequence S=a1⋅a2⋅…⋅aℓ of ℓ elements in Cn, where a repetition of elements is allowed and their order is disregarded. We say that S is a zero-sum sequence if Σi=1ℓai=0 and that S is a zero-sum free sequence if S contains no zero-sum subsequence. In 2000, Gao obtained a construction of all zero-sum free sequences of length n−1−k over Cn for 0≤k≤n3. In this paper, we consider a generalization for a random subset of Cn. Let R=R(Cn,p) be a random subset of Cn obtained by choosing each element in Cn independently with probability p. Let Nn−1−kR be the number of zero-sum free sequences of length n−1−k in R. Also, let Nn−1−k,dR be the number of zero-sum free sequences of length n−1−k having d distinct elements in R. We obtain the expectations of Nn−1−kR and Nn−1−k,dR for 0≤k≤n3 and show that Nn−1−kR and Nn−1−k,dR are asymptotically almost surely (a.a.s.) concentrated around their expectations when k is fixed. Moreover, we provide two ways to compute the expectations using partition numbers and a recurrence formula.