论文标题
在有限循环基团的随机子集中包含的零和不含序列上
On zero-sum free sequences contained in random subsets of finite cyclic groups
论文作者
论文摘要
令$ c_n $为订单$ n $的循环集。 $ c_n $上的长度$ \ ell $的序列$ s $是一个序列$ s = a_1 \ boldsymbol \ cdot a_2 \ boldsymbol \ cdot \ cdot \ cdot \ ldots \ boldsymbol \ cdot a _ {\ ell} $ of c_n $中的$ c_n $中的$ \ ell $ elements and presentiTion和element的命令，并允许均允许的命令。我们说，如果$σ_{i = 1}^{\ ell} a_i = 0 $，则$ s $是零和序列，如果$ s $不包含零和零子序列，则该$ s $是零和免费序列。 令$ r $为$ c_n $的随机子集，通过在$ c_n $中独立选择概率$ p $而获得的每个元素。令$ n^r_ {n-1-k} $为$ r $中的零和长度$ n-1-k $的零和免费序列的数量。另外，令$ n^r_ {n-1-k，d} $为$ d $ dintertion的零和长度$ n-1-k $的零和免费序列的数量。我们获得$ n^r_ {n-1-k} $和$ n^r_ {n-1-k，d} $的期望，对于$ 0 \ leq k \ leq \ leq \ big \ lfloor \ lfloor \ frac {n} {3} {3} \ big big \ rfloor $。我们还显示了$ n^r_ {n-1-k} $和$ n^r_ {n-1-k，d} $时的浓度结果。
Let $C_n$ be a cyclic group of order $n$. A sequence $S$ of length $\ell$ over $C_n$ is a sequence $S = a_1\boldsymbol\cdot a_2\boldsymbol\cdot \ldots\boldsymbol\cdot a_{\ell}$ of $\ell$ 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 $Σ_{i=1}^{\ell} a_i = 0$ 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 $N^R_{n-1-k}$ be the number of zero-sum free sequences of length $n-1-k$ in $R$. Also, let $N^R_{n-1-k,d}$ be the number of zero-sum free sequences of length $n-1-k$ having $d$ distinct elements in $R$. We obtain the expectation of $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ for $0\leq k\leq \big\lfloor \frac{n}{3} \big\rfloor$. We also show a concentration result on $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ when $k$ is fixed.
