What the study found
The authors established a new necessary condition for a tiling of the symmetric group by the identity and all transpositions. They also studied tilings by the set of all transpositions and were led to conjecture that neither set tiles for any value of n.
Why the authors say this matters
The authors state that their condition generalizes a result of Rothaus and Thompson and a result of Nomura. The findings indicate a broader constraint on when these symmetric-group tilings can exist.
What the researchers tested
The researchers studied tilings of a group, meaning that every element must be uniquely written as a product of one element from each of two subsets. In this paper, they focused on the symmetric group and on subsets made from the identity and transpositions, which are swaps of two elements.
What worked and what didn't
They showed that a tiling of the symmetric group by the identity and all transpositions must satisfy a partition-transitivity condition with respect to certain partitions of n. This extends earlier nonexistence results, including the case where n is divisible by a prime number at least 5. They also report that their study of tilings by all transpositions led them to conjecture that neither of the two proposed tilings exists for any n.
What to keep in mind
The abstract gives a necessary condition and a conjecture, not a full classification. It does not provide the detailed proofs or further limitations in the available summary.
Key points
- A new necessary condition was found for tilings of the symmetric group by the identity and all transpositions.
- The condition is partition-transitivity with respect to certain partitions of n.
- The result generalizes earlier work by Rothaus and Thompson and by Nomura.
- The authors also studied tilings by all transpositions and conjecture that neither proposed tiling exists for any n.
- The abstract states an earlier nonexistence result when n is divisible by a prime number at least 5.
Disclosure
- Research title:
- New necessary condition for tilings of the symmetric group
- Authors:
- Teng Fang, Binzhou Xia
- Institutions:
- Soochow University, The University of Melbourne
- Publication date:
- 2026-04-21
- DOI:
- 10.1112/blms.70366
- OpenAlex record:
- View
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