A Note on Ternary Sequences of Strings of 0 and 1

TL;DR

This paper explores permutations of all nonempty subsets of a set of size greater than four, satisfying a specific symmetric difference condition, extending previous conjectures about such sequences.

Contribution

It proves the existence of permutations satisfying the condition for sets with more than four elements, advancing understanding of ternary sequences of binary strings.

Findings

Permutations exist for sets with more than four elements.

The conjecture holds for larger sets beyond the initial case of size two.

Provides new insights into the structure of binary string sequences.

Abstract

B. D. Acharya has conjectured that if $\bigl(A_i: i=1, 2, ..., 2^{|X|}-1\bigr)$ is a permutation of all nonempty subsets of a set $X$ with at least two elements such that for each even positive integer $j<2^{|X|}-1$, $A_{j-1}\triangle A_j\triangle A_{j+1}=\emptyset$, then $|X|=2$. In this article, we show that if the cardinality of a set $X$ is more than four, then a permutation as described above indeed exists.