Shorthand Universal Tori for Permutations: Existence, Symmetry, and Generation of Twori

A de Bruijn sequence packs all $n$-bit binary words into a cycle of length $2^n$. A de Bruijn torus is the two-dimensional analogue in which each word appears exactly once in a rectangular window. Here we consider the natural analogue for permutations using their shorthand representation (i.e., each permutation's final redundant value is omitted from the window). We show that these tori exist when $n = 2m + 1$ is odd and the torus and windows have two rows (i.e., the torus is a "tworus"). These twori can be constructed with a high degree of symmetry. More specifically, there are twori that can be partitioned into $2^{m-1}$ matching blocks where each block contains the same sequence of unordered columns. Furthermore, given one such block we can generate each successive column of a tworus in amortized $\mathcal{O}(1)$-time. We also prove non-existence results for certain sizes of tori and provide algorithms for constructing multiversal cycles (perfect necklaces) of unlabeled binary words.