Periodic balanced binary triangles

TL;DR

This paper proves that balanced binary triangles, with nearly equal numbers of zeroes and ones, exist for all sizes by constructing periodic variants where rows, columns, or diagonals are periodic sequences.

Contribution

It introduces the concept of periodic balanced binary triangles and demonstrates their existence for all positive integer sizes.

Findings

Balanced binary triangles exist for all positive integers n.

Periodic balanced binary triangles can be constructed with specific periodicity properties.

The study extends understanding of combinatorial structures related to Pascal's triangle modulo 2.

Abstract

A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.