We can write $1\in S_n$ non trivialy via different transpositions for $n\geq 4$. For example $1 = (1\;4)(3\;4)(1\;2)(1\;3)(2\;4)(2\;3)$. However if we fix an order of all transpositions such that transpositions in the representation of $1$ have to agree with that order, can we still always find a non trivial representation of $1$? And if no, for what orders can we not find it given $n$?
2026-03-28 08:09:25.1774685365
representation of 1 in the symmetric group by transpositions given a fixed order
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