论文标题
在某些Motzkin对象和$ Q $ -Motzkin号码上的交叉和巢穴
Crossings and nestings over some Motzkin objects and $q$-Motzkin numbers
论文作者
论文摘要
我们根据交叉和巢穴的数量来检查某些Motzkin对象的枚举。关于持续的分数，我们计算并表达了三组交叉和巢穴数量的统计数据的分布，即$ 4321 $避免$ 4321 $ - 避免$ 3412 $避免使用的交易，以及$（321,321,3 \ bar {1} 42} 42} 42）$ - 避免使用的$（321,3 \ bar {1}）。为了获得我们的结果，我们利用Biane的两次审查，仅限于$ 4321 $ - 和3412美元的$ 3412 $ - 避免使用的互动，这些参与的特征是Barnabei等人的特征，而Bivaction和$（321,3 \ bar {1} 42} 42）$ - 避免了定位和Motkkin Paths and。此外，我们操纵所获得的持续分数以获取交叉和巢穴多项式分布的递归公式，因此结果涉及两个新的$ Q $ -Motzkin数字。
We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al.~ and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.~. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.
