Tiling symmetric groups by transpositions
Teng Fang, Binzhou Xia
TL;DR
The paper advances the long-standing question of tiling the symmetric group $S_n$ by the transposition-containing set $T_n$ (and by $T_n^*$, the transpositions without the identity). It introduces a new necessary condition, showing that if a tiling $(T_n,Y)$ exists, then $Y$ must be $\lambda$-transitive for all partitions $\lambda\vdash n$ with nonnegative content $\sum_{x\in D(\lambda)}\xi(x)$, and that certain divisibility constraints on $|T_n|$ and the hook-length product $\lambda_1!\cdots\lambda_\ell!$ must hold. This representation-theoretic approach yields stronger transitivity conclusions (in particular, $Y$ is $k$-transitive for all $k\le n/2$) and a refinement of Rothaus–Thompson’s result, while generalizing Nomura’s multi-transitivity findings; the same framework applied to $T_n^*$ leads to corresponding obstructions. Collectively, these results support the conjecture that neither $T_n$ nor $T_n^*$ tiles $S_n$ for any $n\ge4$, and they illuminate connections to Fourier analysis on $S_n$, eigenvalues of Cayley graphs, and forbidden-intersection problems for permutations. The work thus links group-factorizations with partition-transitivity and provides a structured route to potential resolution via spectral and combinatorial methods.
Abstract
For nonempty subsets $X$ and $Y$ of a group $G$, we say that $(X,Y)$ is a tiling of $G$ if every element of $G$ can be uniquely expressed as $xy$ for some $x\in X$ and $y\in Y$. In 1966, Rothaus and Thompson studied whether the symmetric group $S_n$ with $n\geq3$ admits a tiling $(T_n,Y)$, where $T_n$ consists of the identity and all the transpositions in $S_n$. They showed that no such tiling exists if $1+n(n-1)/2$ is divisible by a prime number at least $\sqrt{n}+2$. In this paper, we establish a new necessary condition for the existence of such a tiling: the subset $Y$ must be partition-transitive with respect to certain partitions of $n$. This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether $S_n$ can be tiled by the set $T_n^*$ of all the transpositions, which finally leads us to conjecture that neither $T_n$ nor $T_n^*$ tiles $S_n$ for any $n\geq4$.