论文标题
关于指定标签最未经污染周期的排列产品
On products of permutations with the most uncontaminated cycles by designated labels
论文作者
论文摘要
自从斯坦利（Stanley）解决Bóna的猜想以来，研究某些标签的分布越来越兴趣。本文涉及到这个方向的问题。令$ d $为集合$ [n] = \ {1,2，\ ldots，n \} $和$ e \ subset [n] $的排列。假设$ d $的$ e $ $ $标签未污染的最大循环数量和$ [n] $的环状排列为$θ$（取决于$ d $和$ e $）。 We prove that for arbitrary $D$ and $E$ with few exceptions, the number of cyclic permutations $γ$ such that $D\circ γ$ has exactly $θ-1$ $E$-label free cycles is at least $1/2$ that of $γ$ for $D\circ γ$ to have $θ$ $E$-label free cycles, where $1/2$ is best possible.也提出了更一般的结果。
There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of Bóna. This paper is concerned with a problem in that direction. Let $D$ be a permutation on the set $[n]=\{1,2,\ldots, n\}$ and $E\subset [n]$. Suppose the maximum possible number of cycles uncontaminated by the $E$-labels in the product of $D$ and a cyclic permutation on $[n]$ is $θ$ (depending on $D$ and $E$). We prove that for arbitrary $D$ and $E$ with few exceptions, the number of cyclic permutations $γ$ such that $D\circ γ$ has exactly $θ-1$ $E$-label free cycles is at least $1/2$ that of $γ$ for $D\circ γ$ to have $θ$ $E$-label free cycles, where $1/2$ is best possible. An even more general result is also conjectured.
