If $\binom{n}{2}$ is even, then can you always express the identity permutation as a product of all distinct transpositions in $S_{n}$?

Asked Dec 26 '23 at 23:42

Active Feb 19 '24 at 00:10

Viewed 111 times

For which $n\geq 2$, is it possible to express the identity permutation as the product of all $\binom{n}{2}$ distinct transpositions in $S_n$? Clearly, we require $\binom{n}{2}$ to be even, which requires $n$ to be $0$ or $1 \pmod{4}$.

We have for $n=4$ that (in cycle notation)

$$ I = (34)(14)(23)(13)(24)(12). $$

For larger $n$, I used a computer to search through possible representations. For $n=5$,

$$ I = (14)(34)(13)(15)(45)(23)(24)(35)(12)(25) $$ and for $n=8$, we have

$$ I = (12)(14)(28)(47)(68)(16)(24)(18)(34)(25)(45)(13)(58)(78)(56)(27)(48)(23)(35)(15)(57)(67)(17)(36)(38)(46)(26)(37). $$

These have no clear pattern. Is this always possible? If it is, it feels like there should be some inductive approach, where we somehow go from a construction for $S_n$ to one for $S_{n+4}$.

edited Feb 19 '24 at 00:10

asked Dec 26 '23 at 23:42

RDL

1

Can every even (odd) permutation be expressed as a product of all distinct transpositions if $\binom n2$ is even (odd)? – bof Dec 27 '23 at 00:51
1

@bof Have a look at this publication – Jean Marie Dec 27 '23 at 06:06
From a friend: https://imgur.com/a/VYfZcvm – Akiva Weinberger Feb 19 '24 at 13:21

If $\binom{n}{2}$ is even, then can you always express the identity permutation as a product of all distinct transpositions in $S_{n}$?

0 Answers0