Counting toroidal binary arrays, II

TL;DR

This paper derives formulas to count toroidal binary arrays with various symmetries, including rotations, reflections, and transpositions, expanding understanding of their combinatorial properties.

Contribution

It introduces explicit formulas for counting toroidal binary arrays under complex symmetry operations, a novel combinatorial enumeration in this context.

Findings

Formulas for counting arrays with row/column rotations and transposition.

Formulas for counting arrays with rotations, reflections, and transposition.

Enhanced understanding of symmetry-based enumeration of binary arrays.

Abstract

We derive formulas for $(i)$ the number of toroidal $n\times n$ binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and $(ii)$ the number of toroidal $n\times n$ binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.