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We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It…
We show that the number of parking functions of length $n$ with zero secondary dinv is equal to the number of ordered cycle decompositions of permutations of $[n]$.
We give a bijective proof of the fact that the number of k-prefixes of minimal factorisations of the n-cycle (1...n) as a product of n-1 transpositions is n^{k-1}\binom{n}{k+1}. Rather than a bijection, we construct a surjection with fibres…
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has…
The \emph{Shi arrangement} is the set of all hyperplanes in $\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \le j < k \le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this…
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
We give a very short proof of the fact that the number of $(a,b)$-parking functions of length $n$ equals $a(a+bn)^{n-1}$. This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general…
A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure…
In this paper, we explore parking distributions on caterpillar trees, focusing on two primary statistics: the number of lucky cars and the frequency with which cars prefer specific parking spaces. We use first-return decomposition to reveal…
Tiered trees were introduced as a combinatorial object for counting absolutely indecomposable representation of certain quivers and torus orbit of certain homogeneous variety. In this paper, we define a bijection between the set of…
In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a $d$-cycle into a product of cycles of prescribed lengths is $d^{r-2}.$ To prove our result, we first define a…
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schr\"oder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over…
A parking function is a function $\pi:[n]\to [n]$ whose $i$th-smallest output is at most $i,$ corresponding to a parking procedure for $n$ cars on a one-way street. We refine this concept by introducing preference-restricted parking…
An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ such that the partition obtained by reordering the values of $f$ fits inside a right triangle with legs of length $m$ and $n$. Recent work by McCammond, Thomas,…
We introduce parking assortments, a generalization of parking functions with cars of assorted lengths. In this setting, there are $n\in\mathbb{N}$ cars of lengths $\mathbf{y}=(y_1,y_2,\ldots,y_n)\in\mathbb{N}^n$ entering a one-way street…
Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their…
Stanley and F\'eray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangular coordinates and gives an…
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the…
We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.
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…