Related papers: Parking functions and triangulation of the associa…
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and…
A~$k$-associahedron is a simplicial complex whose facets, called~$k$-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no~$k+1$ diagonals mutually cross. Such complexes are conjectured for about a decade…
The $k$-associahedron $Ass_k(n)$ is the simplicial complex of $(k+1)$-crossing-free subgraphs of the complete graph with vertices on a circle. Its facets are called $k$-triangulations. We explore the connection of $Ass_k(n)$ with the…
We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the…
Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of triangulations of simplices whose local h-polynomial vanishes. As a first step, we…
We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the…
In this paper, we complete the enumeration of the number of parking functions of length $n$ avoiding, in the sense defined by Qiu and Remmel, a permutation of length 3, answering several questions of Adeniran and Pudwell. Additionally, we…
Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair $(a,b)$, where $a$ is a…
Parking functions are tuples that describe the parking of $M$ cars on a street with $M$ parking spots. In this paper, we define exact $k$-typed parking functions ($k$-TPFs) to be a variant of classical parking functions. We then establish…
We review three realizations of the associahedron that arise as secondary polytopes, from cluster algebras, and as Minkowski sums of simplices, and show that under any choice of parameters, the resulting associahedra are affinely…
Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…
Unit-interval parking functions are subset of parking functions in which cars park at most one spot away from their preferred parking spot. In this paper, we characterize unit-interval parking functions by understanding how they decompose…
Although simplices are trivial from a linear optimization standpoint, the simplex algorithm can exhibit quite complex behavior. In this paper we study the behavior of max-slope pivot rules on (products of) simplices and describe the…
Recall that $\alpha=(a_1,a_2,\ldots,a_n)\in[n]^n$ is a parking function if its nondecreasing rearrangement $\beta=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq i$ for all $1\leq i\leq n$. In this article, we study parking functions based on…
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize…
For any irreducible real reflection group $W$ with Coxeter number $h$, Armstrong, Reiner, and the author introduced a pair of $W \times \ZZ_h$-modules which deserve to be called {\sf $W$-parking spaces} which generalize the type A notion of…
We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry,…
We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson,…
We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…
Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking…