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

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