Related papers: Permutation Invariant Parking Assortments

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their…

We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…

Warning. The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up!

Stirling permutations are parking functions, and we investigate two parking function statistics in the context of these objects: lucky cars and displacement. Among our results, we consider two extreme cases: extremely lucky Stirling…

Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a…

Let $W$ be an irreducible finite complex reflection group acting on a complex vector space $V$. For a positive integer $k$, we consider a class function $\varphi_k$ given by $\varphi_k(w) = k^{\dim V^w}$ for $w \in W$, where $V^w$ is the…

Multiplicative invariance is a well-studied property of subsets of the unit interval. The theory in the complex plane is less developed. This paper introduces an analogous definition for multiplicative invariance in the complex plane…

All over the world, especially in the university environment, planning managers and traffic engineers are constantly faced with the problem of inadequate allocation of car parking spaces to demanded users. Users could either prefer reserved…

Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each…

The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$.…

A Parity Alternating Permutation of the set $[n] = \{1, 2,\ldots, n\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the…

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations…

Let $t$ be a permutation (that shall play the role of the {\em text}) on $[n]$ and a pattern $p$ be a sequence of $m$ distinct integer(s) of $[n]$, $m\leq n$. The pattern $p$ occurs in $t$ in position $i$ if and only if $p_1... p_m$ is…

In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval…

Recently, the authors extended the notion of parking functions to parking sequences, which include cars of different sizes, and proved a product formula for the number of such sequences. We here give a refinement of that result involving…

We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a…

This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking…

This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number…

In a scenario of growing usage of park-and-ride facilities, understanding and predicting car park occupancy is becoming increasingly important. This study presents a model that effectively captures the occupancy patterns of park-and-ride…

Consider the following simple parking process on $\Lambda_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $\Lambda_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied,…