On Balanced Colorings of the n-Cube

TL;DR

This paper proves a conjecture about the symmetry and unimodality of the number of balanced 2-colorings of the n-cube and explores their log-concavity properties, providing new insights into combinatorial coloring patterns.

Contribution

It proves the symmetry and unimodality conjecture for the sequence of balanced colorings and proposes a new conjecture on their log-concavity, supported by probabilistic methods.

Findings

Proof of symmetry and unimodality of B_{n,2k} sequence.

Conjecture on log-concavity of B_{n,2k} for fixed k.

Validation of log-concavity for large n using probabilistic methods.

Abstract

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored n-cube is said to be balanced if its center of mass coincides with its geometric center. Let $B_{n,2k}$ be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read and Robinson conjectured that for $n\geq 1$, the sequence $\{B_{n,2k}\}_{k=0, 1 ... 2^{n-1}}$ is symmetric and unimodal. We give a proof of this conjecture. We also propose a conjecture on the log-concavity of $B_{n,2k}$ for fixed k, and by probabilistic method we show that it holds when n is sufficiently large.