An atomic model of the graph of permutations linked by transpositions
The authors study the Cayley graph for the symmetric group, generated by transpositions. (This graph has relevance in campanology as well, but the authors do not investigate this connection.) They note that minimal product length in generators is determined by cycle structure; thus a partition of permutations by minimal word length can be refined into a partition by conjugacy classes. The author study (using language borrowed from the theory of atoms) such ``shells'' of elements of same minimal word length, as well as connections between different shells. Using standard ``energy minimization'' methods, They study how these graphs could be embedded, for \(n=3,4\) into \(\mathbb{R}^3\) in pleasing configurations and illustrate these through images. (It is noted that this approach performs less well in degree 5.)
scientific article; zbMATH DE number 7606353