Related papers: Parking Functions and Descent Algebras
Functors from (co)operads to bialgebras relate Hopf algebras that occur in renormalisation to operads, which simplifies the proof of the Hopf algebra axioms, and induces a characterisation of the corresponding group of characters and Lie…
We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and…
The syntactic Merge operation of the Minimalist Program in linguistics can be described mathematically in terms of Hopf algebras, with a formalism similar to the one arising in the physics of renormalization. This mathematical formulation…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines…
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…
In this paper, we propose a novel hash learning approach that has the following main distinguishing features, when compared to past frameworks. First, the codewords are utilized in the Hamming space as ancillary techniques to accomplish its…
The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product…
The shuffle conjecture expresses a relationship between parking functions, diagonal harmonics, and the Bergeron-Garsia $\nabla$ operator. Recent conjectures about a family of modified Hall-Littlewood operators made by Haglund, Morse, and…
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…
The downward closure of a language $L$ of words is the set of all (not necessarily contiguous) subwords of members of $L$. It is well known that the downward closure of any language is regular. Although the downward closure seems to be a…
This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedler's dual.
The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among…
We briefly review the Hopf algebra structure arising in the renormalization of quantum field theories. We construct the Hopf algebra explicitly for a simple toy model and show how renormalization is achieved for this particular model.
Let $W$ be an irreducible real reflection group. Armstrong, Reiner, and the author presented a model for parking functions attached to W and made three increasingly strong conjectures about these objects. The author generalized these…
The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in…
Let $\mathbf D$ be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On $\BZ\mathbf D$ we define a product and a coproduct, together with an internal product, that is,…
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…
Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category…
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…