# [r-t] Court Bob Minor

Richard Smith richard at ex-parrot.com
Tue Sep 19 20:22:10 UTC 2017

```Robin Woolley wrote:

> Perhaps we are at cross purposes or something, but all transpositions of
> Reverse Bob Doubles are of even nature.
>
> e.g., row 5 = 45231 and 54231. Transposition = 21345 which looks even to
> me, given that I mean that T^2=e.

But that's not what an even transposition is.  An even
transposition is a perfectly standard term, both in ringing
or in mathematics.  It is transposition is one which maps
even parity rows to even parity rows, and odd parity rows to
odd parity rows.  21345 is not an even transposition: it's
about the most obvious possible example of an odd
transposition as it swaps precisely one pair.

If what you mean is that the transposition squares to the
identity (but is not the identity itself), that's an
entirely different concept.  Such transpositions are said to
have order 2, because n=2 is the lowest n such that t^n = e
where t is the transposition and e the identity.  Or we can
say they're self-inverse (though strictly this applies to
the identity too).  In the mathematical literature such
transpositions are sometimes referred to as involutions.

But this has nothing to do with whether the transposition is
odd or even.  Whether a transposition is odd or even is
orthogonal to whether it is a involution.  21435 is an even
involution, and 21345 is an odd involution; 23145 is an even
transposition that is not an involution, while 23154 is an
odd transposition that is not an involution.

If you're going to make up your own terminology, do try not
to assign new and contradictory meanings to terminology
that's already in common used; or if you must do so, at
least warn people how you're using the terms.

RAS

```