论文标题
浅排列是未链接排列的
The shallow permutations are the unlinked permutations
论文作者
论文摘要
DiConis和Graham研究了一个名为“总位移”的对称组中的距离的量度，并表明它以长度和反射长度的总和为界。他们要求描述该界限是平等的排列。我们将这些称为浅排列。 Cornwell和McNew最近将置换列表的循环图解释为结图，并研究了相应链接是UNLINK的一组排列。我们显示浅排列完全是未连接的排列。由于Cornwell和McNew给出了计数未链接排列的生成函数，这给出了生成函数计数浅排列。
Diaconis and Graham studied a measure of distance from the identity in the symmetric group called total displacement and showed that it is bounded below by the sum of length and reflection length. They asked for a characterization of the permutations where this bound is an equality; we call these the shallow permutations. Cornwell and McNew recently interpreted the cycle diagram of a permutation as a knot diagram and studied the set of permutations for which the corresponding link is an unlink. We show the shallow permutations are precisely the unlinked permutations. As Cornwell and McNew give a generating function counting unlinked permutations, this gives a generating function counting shallow permutations.
