On Flattened Parking Functions

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. We show that the set of flattened parking functions with one run is enumerated by the Catalan numbers, as they are in bijection with nondecreasing parking functions. 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. As a corollary, we establish that $\textbf{1}_r$-insertion flattened parking functions are enumerated by the $r$-Bell numbers. We give the closed differential form for the exponential generating functions of these enumerations. Our results generalize the work of Nabawanda, Rakotondrajao, Bamunoba and Beyene, Mantaci on flattened partitions.