[r-t] Parity
Ben Willetts
ben at benjw.org.uk
Tue Jun 19 17:27:04 UTC 2007
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
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