[r-t] Superpermutations

[Simon Humphrey]

I had a brief look at superpermutations some months ago, hoping to find ringable systems to produce minimum length ones.
The optimal length-33 superpermutation on 4 bells could be rung if bells are allowed to be omitted from a row, maintaining a strict 4-bell cartwheel rhythm by substituting a gap for each missing bell. Doing this means the backstrokes and handstrokes get mixed up, of course, as in infinite hunting, and the bells don't all ring an equal number of times.

1	2	3	4
1	2	3	
1	4	2	3
1	2	4	3
1	2		
1	3	4	2
1	3	2	4
1	3	2	
1	4	3	2
1			

The treble has an easy time, leaving gaps for 3 bells between every stroke.
The gap patterns for the other bells are
2 : 3,4,2,3,5,2,3,4
3 : 3,4,3,5,3,3,4
4 : 5,4,7,4,5

I can't see us trying this on a practice night any time soon though.

It's annoying that the gap patterns for 2 and 3 are asymmetrical.  Are there any other optimal length superpermutations on 4, that give symmetrical patterns for all 4 bells?

Relaxing the minimal length requirement and basing method constructions on proper rows with each bell sounding exactly once, as Philip demonstrates, would obviously provide much more scope.

SH