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 $\mathrm{\ell }$ over $C_n$ is a sequence $S = a_1\boldsymbol\cdot a_2\boldsymbol\cdot \ldots\boldsymbol\cdot a_{\ell}$of$\ell$elementsin$C_n$, where a repetition of elements is allowed and their order is disregarded. We say that $S$ is a zero-sum sequence if ${\mathrm{\Sigma }}_{i=1}^{\mathrm{\ell }}{a}_i=0$ and that $S$ is a zero-sum free sequence if $S$ contains no zero-sum subsequence. Let $R$ 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 expectation of $N_{n-1-k}^R$ and $N_{n-1-k,d}^R$ for $0\le k\le \lfloor \frac{n}{3}\rfloor$. We also show a concentration result on $N_{n-1-k}^R$ and $N_{n-1-k,d}^R$ when $k$ is fixed.