论文标题
在扁平的停车功能上
On Flattened Parking Functions
论文作者
论文摘要
如果上升的最大链（称为运行）的主要链条越来越多，则置于$ n $的长度$ n $的排列称为平坦的分区。我们类似地定义了扁平的停车功能：停车功能的一个子集，该功能的最大链链的主要条款（也称为跑步）的顺序较弱。对于$ n \ leq 8 $（最多有四次运行），我们提供了扁平停车功能数量的数据，并且为其一般的枚举提供公式仍然是一个开放的问题。然后，我们专门研究称为$ \ Mathcal {s} $ - 插入扁平的停车功能的一部分。这些是通过将所有数字$ \ MATHCAL {s} $插入其元素属于$ [n] = \ {1,2，\ ldots，n \} $中的所有数字来获得的。我们提供$ \ MATHCAL {S} $ - 插入扁平的停车功能与$ \ Mathcal {s}'$ - 插入扁平的停车功能之间的将进行的，其中$ \ MATHCAL {S} $和$ \ MATHCAL {s}'$具有某些关系。然后，我们进一步专注于$ \ MATHCAL {s} = \ textbf {1} _r $，带有$ r $的多机，我们在$ \ textbf {1} _r $ insertion flatted停车功能和$ [n+r] $的$ r $ r $ r $ r $ r $ ressets中建立了两次射击。
A permutation of length $n$ is called a flattened partition if the leading terms of maximal chains of ascents (called runs) are in increasing order. We analogously define flattened parking functions: a subset of parking functions for which the leading terms of maximal chains of weak ascents (also called runs) are in weakly increasing order. For $n\leq 8$, where there are at most four runs, we give data for the number of flattened parking functions, and it remains an open problem to give formulas for their enumeration in general. We then specialize to a subset of flattened parking functions that we call $\mathcal{S}$-insertion flattened parking functions. These are obtained by inserting all numbers of a multiset $ \mathcal{S}$ whose elements are in $[n]=\{1,2,\ldots,n\}$, into a permutation of $[n]$ and checking that the result is flattened. We provide bijections between $\mathcal{S}$-insertion flattened parking functions and $\mathcal{S}'$-insertion flattened parking functions, where $\mathcal{S}$ and $\mathcal{S}'$ have certain relations. We then further specialize to the case $\mathcal{S}=\textbf{1}_r$, the multiset with $r$ ones, and we establish a bijection between $\textbf{1}_r$-insertion flattened parking functions and set partitions of $[n+r]$ with the first $r$ integers in different subsets.
