Narayana numbers in "explicit sufficient invariants for an interacting particle system ( by Itoh, Mallows, and Shepp)"

TL;DR

This paper studies an interacting particle system on star graphs, revealing infinitely many martingale invariants linked to Narayana numbers, which help determine the system's limiting distribution and death state probabilities.

Contribution

It introduces a novel connection between Narayana numbers and martingale invariants in an interacting particle system on star graphs.

Findings

Martingale invariants are homogeneous polynomials with Narayana number coefficients.

The Narayana number identity is used to compute death state probabilities.

The system's limiting distribution can be derived using these invariants.

Abstract

We consider an interacting particle system on star graphs. As in the case of the Kdv equation, we have infinitely many invariants ( here, martingale invariants). It enables us to obtain the limiting distribution of the Markov chain. Each of the martingale invariants is a homogeneous polynomial with coefficients of Narayana numbers.The identity for the enumeration of plane unlabeled trees, which gives Narayana numbers, becomes the key identity to obtain the probability of death states by a change of variables.