相关论文: Enumerating Pattern Avoiding Parking Functions

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare…

In this paper we study the enumeration and the construction of particular binary words avoiding the pattern $1^{j+1}0^j$. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule,…

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of…

This paper provides an exploration of parking functions, a classical combinatorial object. We present two viewpoints on their structure and properties: through poset of noncrossing partitions and polytopes.

Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…

We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions.…

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the…

In a recent paper by Kitaev and Remmel, several formulas for the number of words of length n avoiding some generalized patterns were established. Each time the obtained function of n had been found in Sloane's Encyclopedia as the number of…

The number of inversion sequences avoiding two patterns $101$ and $102$ is known to be the same as the number of permutations avoiding three patterns $2341$, $2431$, and $3241$. This sequence also counts the number of Schr\"{o}der paths…

We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. 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…

Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be…

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…

We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of…

A Catalan word of length $n$ that avoids the pattern $(\geq, \geq)$ is a sequence $w=w_1\cdots w_n$ with $w_1=0$ and $0\leq w_i\leq w_{i-1}+1$ for all $i$, while ensuring that no subsequence satisfies $w_i \geq w_{i+1}\geq w_{i+2}$ for…

In the set of all patterns in $S_n$, it is clear that each k-pattern occurs equally often. If we instead restrict to the class of permutations avoiding a specific pattern, the situation quickly becomes more interesting. Mikl\'os B\'ona…

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This…

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average…

This work is a continuation of some recent articles presenting enumerative results for Catalan words avoiding one or a pair of consecutive or classical patterns of length $3$. More precisely, we provide systematically the bivariate…

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…