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

Let C_n be a cyclic group of order n. A sequence S of length ℓ over C_n is a sequence S=a₁⋅a₂⋅…⋅a_ℓ of ℓ elements in C_n, where a repetition of elements is allowed and their order is disregarded. We say that S is a zero-sum sequence if Σ_i=1^ℓa_i=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 C_n for [Formula presented]. In this paper, we consider a generalization for a random subset of C_n. Let R=R(C_n,p) be a random subset of C_n obtained by choosing each element in C_n independently with probability p. Let N_n−1−k^R be the number of zero-sum free sequences of length n−1−k in R. Also, let N_n−1−k,d^R be the number of zero-sum free sequences of length n−1−k having d distinct elements in R. We obtain the expectations of N_n−1−k^R and N_n−1−k,d^R for [Formula presented] and show that N_n−1−k^R and N_n−1−k,d^R 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.