On the Existence and Uniqueness of Symmetric Structures Generating Complete Ordered Pairs

Abstract: This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike other combinatorial constructions, these structures are defined solely by the relative positions of the elements, not their concrete values. Two structures are considered equivalent if one can be obtained from the other through row permutation and/or global relabeling. Under this notion, it is shown that for $n=4$ there exists a single non-trivial structure, and for $n=6$ there are exactly two non-equivalent ones. Harmonic matrices are constructed using specially designed permutators whose properties guarantee symmetry and complete coverage. Their internal hierarchy, extensibility, and rarity within the space of permutations are analyzed. Furthermore, it is demonstrated how these matrices can be used to generate valid Sudoku boards deterministically, without random methods or post-validation. These properties open new perspectives in combinatorics, algorithm design, and systems based on positional encoding. Notably, these permutators enable the construction of harmonic matrices for arbitrary even values of $n$, ensuring the universal scalability of the method.