论文标题
连接$ k $ naples停车功能和通过涉及的停车功能阻塞
Connecting $k$-Naples parking functions and obstructed parking functions via involutions
论文作者
论文摘要
$ N $汽车试图在一个单向街道上停车的$ n $停车位的停车功能是经典定义的，那里的汽车只开车前进。随后，停车功能已通过各种方式推广，包括允许汽车向后行驶。 $ k $ naples停车功能的套装$ pf_ {n，k} $具有汽车，可以在向前驾驶之前最多向后驱动$ k $步骤。 $ | pf_ {n，k} | $的递归公式，尽管为$ | pf_ {n，k} | $得出了一个封闭公式似乎很困难。此外，$ pf_ {n，k} $的重要子集$ b_ {n，k} $称为包含$ k $ - naples停车功能，具有非主体证明，具有与$ k $相同的经典停车功能的$ pf_n $的基数。 在本文中，我们在$ M $ CARS和$ n $停车位的更一般环境中研究$ k $ naples停车功能，对于任何$ m \ leq n $。我们使用各种停车功能来确定包含的$ k $ naples停车功能与经典停车功能之间的两次射击，从中可以推断出两组具有相同数量的关系。然后，我们扩展了这项两次射击，以将$ K $ naples停车功能注入一组阻塞的停车功能，为前一组的基数提供了上限。
Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.
