Existence Classification of Shorthand Universal Tori

Determine for which values of the torus dimensions and window dimensions 8R,C;r,c9 shorthand universal tori of permutations exist, where an R-by-C toroidal grid contains every (n)-permutation exactly once in an r-by-c window with rc=n and RC=n!.

Background

The paper introduces shorthand universal tori of permutations as two-dimensional analogues of shorthand universal cycles. An R-by-C toroidal grid is required to contain every word in the set of (n)-permutations of [n] exactly once in row-major order within an r-by-c window, with the window area equal to n and the torus area equal to n!.

The paper proves existence for the two-row case in which n=2m+1 is odd, the torus has dimensions 2-by-(n!/2), and the windows have dimensions 2-by-m. It also establishes non-existence for two specific 2-by-2-window sizes when n=5, namely 3-by-40 and 5-by-24, while noting that the available computational results do not reveal a general existence pattern. The open problem asks for a complete characterization of all parameter quadruples (R,C;r,c) for which such shorthand universal tori exist.