# Analytic combinatorics of coordination numbers of cubic lattices

## TL;DR

This paper explores the mathematical properties of coordination numbers in cubic lattices, analyzing their positivity, zero distribution, asymptotic behavior, and concavity/convexity to deepen understanding of their combinatorial structure.

## Contribution

It provides a comprehensive analytic study of coordination numbers in cubic lattices, including properties like total positivity and asymptotic normality, which were not previously established.

## Key findings

- Coordination matrices are totally positive.
- Coordination polynomials' zeros are distributed in specific patterns.
- Coordination numbers exhibit asymptotic normality.

## Abstract

We investigate coordination numbers of the cubic lattices with emphases on their analytic behaviors, including the total positivity of the coordination matrices, the distribution of zeros of the coordination polynomials, the asymptotic normality of the coefficients of the coordination polynomials, the log-concavity and the log-convexity of the coordination numbers.