论文标题
通过排列计算$ k $ naples停车功能和$ k $ naples区域统计数据
Counting $k$-Naples parking functions through permutations and the $k$-Naples area statistic
论文作者
论文摘要
我们记得，$ k $ naples停车功能长度为$ n $（停车功能的概括）是通过要求找到其首选地点占用的汽车必须首先备份一个位置（最多可$ k $ spots），然后才能向前推进街上。请注意，停车功能是$ k $至$ 0 $的专业化。对于固定的$ 0 \ leq k \ leq n-1 $，我们定义一个函数$φ_K$，该函数将$ k $ - naples停车功能映射到置换，以表示其汽车公园的订单。通过列举地图纤维的尺寸$φ_K$，我们为$ k $ naples停车的数量提供了一个新的公式，作为长度$ n $的排列的总和。 我们指出，与以前已知的Christensen等人[CHJ+20]相比，我们列举$ k $ naples停车功能的公式不是递归的。它可以表示为特定排列子序列长度的产物，其对$ k = 0 $的专业化提供了一种新的方法来描述长度$ n $的停车功能数量。我们给出了地图$φ_0$的纤维大小的公式，并为其相应的对数生成函数提供了复发关系。此外，我们将公式的$ q $ - analog与一个新的统计量相关联，我们表示$ \ texttt {abreat} _k $，并称呼$ k $ -naples apraples able afor abor afor able-naples apectriatiation，$ k = 0 $的专业化给出了$ \ \ \ \ \ \ \ \ \ \ \ texttttt {abreatt {abreation} $ satistical ot factions functions。
We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before proceeding forward down the street. Note that the parking functions are the specialization of $k$ to $0$. For a fixed $0\leq k\leq n-1$, we define a function $φ_k$ which maps a $k$-Naples parking function to the permutation denoting the order in which its cars park. By enumerating the sizes of the fibers of the map $φ_k$ we give a new formula for the number of $k$-Naples parking functions as a sum over the permutations of length $n$. We remark that our formula for enumerating $k$-Naples parking functions is not recursive, in contrast to the previously known formula of Christensen et al [CHJ+20]. It can be expressed as the product of the lengths of particular subsequences of permutations, and its specialization to $k=0$ gives a new way to describe the number of parking functions of length $n$. We give a formula for the sizes of the fibers of the map $φ_0$, and we provide a recurrence relation for its corresponding logarithmic generating function. Furthermore, we relate the $q$-analog of our formula to a new statistic that we denote $\texttt{area}_k$ and call the $k$-Naples area statistic, the specialization of which to $k=0$ gives the $\texttt{area}$ statistic on parking functions.
