What the study found
The authors establish a new necessary condition for when the symmetric group can be tiled by the identity and transpositions, namely that the subset must be partition-transitive with respect to certain partitions. They also study tiling by the set of all transpositions and are led to conjecture that neither of the two tiling forms exists for any n.
Why the authors say this matters
The study suggests that the new condition extends earlier results by Rothaus and Thompson and by Nomura. The authors present it as a generalization of those prior findings.
What the researchers tested
The paper examines tilings of the symmetric group, meaning unique representations of each group element as a product of one element from each of two subsets. The authors focus on the case where one subset is the identity together with all transpositions, and they also consider tiling by the full set of transpositions.
What worked and what didn't
A new necessary condition was obtained: the relevant subset must be partition-transitive with respect to certain partitions of the symmetric group. This condition generalizes earlier results. The paper also reports that studying tiling by all transpositions leads the authors to conjecture that neither tiling form exists for any n.
What to keep in mind
The abstract states a necessary condition and a conjecture, not a complete classification. It does not describe any limitations beyond the specific tiling settings considered.
Key points
- The paper gives a new necessary condition for tilings of the symmetric group by the identity and transpositions.
- The condition is that the subset must be partition-transitive with respect to certain partitions.
- The result generalizes earlier work by Rothaus and Thompson, and by Nomura.
- The authors also examine tiling by all transpositions.
- From that study, they conjecture that neither tiling form exists for any n.
Disclosure
- Research title:
- New condition for tilings of the symmetric group
- Authors:
- Teng Fang, Binzhou Xia
- Institutions:
- Soochow University, The University of Melbourne
- Publication date:
- 2026-04-21
- OpenAlex record:
- View
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