Parking Functions and Descent Algebras

Parking Functions and Descent Algebras

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 contains the descent algebra as a left ideal.

Permutation Group Algebras and Parking Functions

Hopf Algebras and Dendriform Structures Arising from Parking Functions

We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n + 1) in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schröder numbers (plane trees)...

Genomes and Parking Functions

This paper will focus on rearrangements of genomes containing linear chromosomes. The goal is to be able to be able to enumerate thee rearrangements. This will be done first by determining the number of operations required to rearrange one genome into another. Then if the order in which these operations occur can be altered, the resulting effects on the enumeration will be determined. We will b...

Generalized Parking Functions, Descent Numbers, and Chain Polytopes of Ribbon Posets

We consider the inversion enumerator In(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = −1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula In(−1) = En, the number of alternating perm...