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Jean-Baptiste Priez

Publications and source records attributed to Jean-Baptiste Priez.

4 recordsLinked to original sources
arXiv2015

Non-commutative Frobenius characteristic of generalized parking functions -- Application to enumeration

We give a recursive definition of generalized parking function that allows us to view them as a species. From there we compute a non-commutative characteristic of the generalized parking function module, and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non-commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.

math.CO↗
arXiv2015

Bijection: Parking-like structures and Tree-like structures

We recall the occupancy problem introduced by Konheim & Weiss in 1966 and we consider parking functions as hash maps. Each car $c_i$ prefers parking space $p_i$ (the hash map $c_i \mapsto p_i$ with $c_i$ is a key and $p_i$ an index into an array), if $p_i$ is occupied then $c_i$ the next available parking space (the hash table implementation using an open addressing strategy). This paper considers some others hash table implementations like hash tables with linked lists (with parking functions as hash maps). Using the Species Theory, we enumerate by Lagrange inversion those hash tables structures via a bijection with tree-like structures. This bijection provides a generalization of the Foata-Riordan bijection between parking functions and (forests of) rooted trees. Finally we show the number of hash tables with linked lists on a set of keys of cardinality $n$ is $n!C_n$, so the number of labeled binary trees with $n$ nodes.

math.CO↗
arXiv2013

A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities

In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d'algèbres de Hopf combinatoires à partir d'une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l'utilise pour démontrer une formule des équerres sur ces arbres.

math.CO↗