A generalization of carries process and Eulerian numbers

TL;DR

This paper generalizes the carries process and Eulerian numbers by extending Holte's matrix to non-standard and negative base numeration systems, providing explicit stationary distributions and revealing new combinatorial properties.

Contribution

It introduces a broad generalization of the carries process, connecting it with generalized Eulerian and MacMahon numbers, and explores properties in negative base systems.

Findings

Explicit stationary distributions for generalized carries processes

Extension of properties to negative base numeration systems

Connection between carries process and generalized Eulerian numbers

Abstract

We study a generalization of Holte's amazing matrix, the transition probability matrix of the Markov chains of the 'carries' in a non-standard numeration system. The stationary distributions are explicitly described by the numbers which can be regarded as a generalization of the Eulerian numbers and the MacMahon numbers. We also show that similar properties hold even for the numeration systems with the negative bases.