[r-t] Parity

[Ben Willetts]

Leigh Simpson:
> If we write the perms on 4 in lexicographic order, the
> parities follow a simple alternating pattern:
> 1234 +
> 1243 -
> 1324 -
> 1342 +
> 1423 +
> 1432 -
> 2134 -
> Should this be obvious?

Yes.

Rounds is positive, by definition.  After listing rounds, we have exhausted 
the possibilities with the first four bells fixed (obviously!).  We have 
also exhausted the possibilities with the first three bells fixed, so the 
first bell to vary is the third in the row.  Because of the way we are 
listing the perms, we exchange the third and fourth bells in the row, which 
results in a negative row.

We've now exhausted the possibilities with the first two bells fixed, so the 
next to vary is the second in the row, and, again, we simply swap the second 
and third from 1234, so again we have a negative row.  Now we have bells 
available to swap in 3/4 again, so a positive row results from these.

For the next row, we can't simply swap a pair, so we rotate the 'back 3' --  
keeping the positive parity from rounds.  Proceed as before.

Does that make sense?  It does to me, but I'm not sure I've explained it 
very well!

> Does it extend to higher numbers of bells?

Yes.

12345 +
12354 -
12435 -
12453 +
12534 +
12543 -
13245 -
13254 +
13425 +
etc

123456 +
123465 -
123546 -
123564 +
123645 +
123654 -
124356 -
etc

It extends in the same way because of the order in which we are listing the 
permutations.

Ben