** **## Bumping sequences and multispecies juggling

**Ayyer A.** (

*Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India*)

**Bouttier J.** (

*CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France*)

**Corteel S.** (

*LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13*)

**Linusson S.** (

*Department of Mathematics, KTH-Royal Institute of Technology, SE-100 44, Stockholm, Sweden*)

**Nunzi F.** (

*LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13, France*)

**Abstract:**

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.

**Année de publication ** : 2018

**Revue ** : Advances in Applied Mathematics

**98** 100-126 (2018)

**DOI ** :

10.1016/j.aam.2018.03.001**Preprint ** :

arXiv:1504.02688 **Lien ** :

https://www.sciencedirect.com/science/article/pii/S0196885818300332 **Keywords ** : Markov chains; Combinatorics; Juggling

**Langue ** : Anglais

** Fichier(s) à télécharger : ** publi.pdf