# Integrability properties of Motzkin polynomials

## TL;DR

This paper explores a Hamiltonian system related to matrix scalar field theory, revealing that its conserved currents follow a pattern governed by Motzkin polynomials, linking integrability to combinatorial paths.

## Contribution

It uncovers a novel connection between conserved currents in a generalized shock-wave equation and Motzkin polynomials, highlighting a combinatorial structure in integrability.

## Key findings

- Conserved currents follow a pattern governed by Motzkin polynomial coefficients.
- Each integral of motion corresponds to a path on a unit lattice.
- The system generalizes known integrable shock-wave equations.

## Abstract

We consider a Hamiltonian system which has its origin in a generalization of exact renormalization group flow of matrix scalar field theory and describes a non-linear generalization of the shock-wave equation that is known to be integrable. Analyzing conserved currents of the system the letter shows, that these follow a nice pattern governed by coefficients of Motzkin polynomials, where each integral of motion corresponds to a path on a unit lattice.