From pje24 at cantab.net Wed May 22 09:56:13 2019 From: pje24 at cantab.net (Philip Earis) Date: Wed, 22 May 2019 09:56:13 +0100 (BST) Subject: [r-t] Superpermutations Message-ID: <37808.198.176.82.34.1558515373.squirrel@www.cantab.net> Things have been rather quiet on here recently, so I'd like to rouse everyone from their slumbers. Inspired by a recent mention on the Boston change ringers mailing list, I've become aware of superpermutations. As the wikipedia page (https://en.wikipedia.org/wiki/Superpermutation) explains, a superpermutation on n symbols is a string that contains each permutation of n symbols as a substring. To think about it in more of a change ringing way, it's the shortest possible way of having all n! permutations without having to do this in n! discrete rows. There seems to have been lots of recent activity in the field of superpermutations, both from a combinatorial mathematics perspective and also at applying this to bell ringing. See for example these three links: http://www.gregegan.net/SCIENCE/Superpermutations/Superpermutations.html#NEWS2 https://arxiv.org/abs/1408.5108 https://www.youtube.com/watch?v=_tpNuulTeSQ In the youtube video and comments beneath this, there is various discussion about applying superpermutations to change ringing. A chap called Kurt Ludwick points out: === If we stick to standard change ringing rules, superpermutations can be embedded in change ringing patterns for 3 or 4 bells. On 3 bells: the optimal (length 9) superpermutation 312313213 is embedded in the following method (starting after the leading 1 2): 123 / 123 / 132 / 132 On 4 bells: the optimal (length 33) superpermutation cannot be rung, since it contains a "121"-type subsequence halfway through. However, the following method contains a superpermutation of length 34 (also starting after the leading 1 2): 1234 / 1234 /1324 / 3124 / 3214 / 2314 / 2134 / 2143 / 2413 This is the minimal change-ringable (?) superpermutation on 4 bells. For more than 4 bells: superpermutation change ringing methods are possible if we relax the definition of "superpermutation" to allow for some repeated permutations (i.e., each occurs at least once, rather than exactly once). The shortest such superpermutation I've found for 5 bells has length 160, embedded within a method of 33 changes. I suspect this may be minimal for 5 bells, but can't say for sure === Perhaps more excitingly, on 27 February this year Robin Houston and Greg Egan found a superpermutation for n=7 of length 5906, breaking the previous record. This built on their finding a superpermutation of length 872 for n=6. The n=7 superpermutation is copied below. You can listen to this being performed (on a piano) at https://www.youtube.com/watch?v=n4WpyNMfQ2Q. Lots of avenues to explore here. Can anyone arrange "superpermutation blocks" for 5 or 6 bells, both with and without jump changes? 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blueyonder.co.uk (Glenn Taylor) Date: Wed, 22 May 2019 12:08:13 +0100 Subject: [r-t] Superpermutations In-Reply-To: <37808.198.176.82.34.1558515373.squirrel@www.cantab.net> References: <37808.198.176.82.34.1558515373.squirrel@www.cantab.net> Message-ID: <000001d5108e$a65fbbf0$f31f33d0$@blueyonder.co.uk> Thanks, Philip. This is not a topic that I am familiar with, but it reaches towards the answer to a question that came to my mind quite some time ago: what is the shortest string of digits that needs to be entered into a 4-digit burglar alarm to test all the possibilities without trying them all in turn. Unfortunately n>5 (q.v. Wikipedia article)... Cheers, Glenn -----Original Message----- From: ringing-theory [mailto:ringing-theory-bounces at bellringers.org] On Behalf Of Philip Earis Sent: 22 May 2019 09:56 To: ringing-theory at bellringers.org Subject: [r-t] Superpermutations Things have been rather quiet on here recently, so I'd like to rouse everyone from their slumbers. Inspired by a recent mention on the Boston change ringers mailing list, I've become aware of superpermutations. As the wikipedia page (https://en.wikipedia.org/wiki/Superpermutation) explains, a superpermutation on n symbols is a string that contains each permutation of n symbols as a substring. To think about it in more of a change ringing way, it's the shortest possible way of having all n! permutations without having to do this in n! discrete rows. There seems to have been lots of recent activity in the field of superpermutations, both from a combinatorial mathematics perspective and also at applying this to bell ringing. See for example these three links: http://www.gregegan.net/SCIENCE/Superpermutations/Superpermutations.html#NEWS2 https://arxiv.org/abs/1408.5108 https://www.youtube.com/watch?v=_tpNuulTeSQ In the youtube video and comments beneath this, there is various discussion about applying superpermutations to change ringing. A chap called Kurt Ludwick points out: === If we stick to standard change ringing rules, superpermutations can be embedded in change ringing patterns for 3 or 4 bells. On 3 bells: the optimal (length 9) superpermutation 312313213 is embedded in the following method (starting after the leading 1 2): 123 / 123 / 132 / 132 On 4 bells: the optimal (length 33) superpermutation cannot be rung, since it contains a "121"-type subsequence halfway through. However, the following method contains a superpermutation of length 34 (also starting after the leading 1 2): 1234 / 1234 /1324 / 3124 / 3214 / 2314 / 2134 / 2143 / 2413 This is the minimal change-ringable (?) superpermutation on 4 bells. For more than 4 bells: superpermutation change ringing methods are possible if we relax the definition of "superpermutation" to allow for some repeated permutations (i.e., each occurs at least once, rather than exactly once). The shortest such superpermutation I've found for 5 bells has length 160, embedded within a method of 33 changes. I suspect this may be minimal for 5 bells, but can't say for sure === Perhaps more excitingly, on 27 February this year Robin Houston and Greg Egan found a superpermutation for n=7 of length 5906, breaking the previous record. This built on their finding a superpermutation of length 872 for n=6. The n=7 superpermutation is copied below. You can listen to this being performed (on a piano) at https://www.youtube.com/watch?v=n4WpyNMfQ2Q. Lots of avenues to explore here. Can anyone arrange "superpermutation blocks" for 5 or 6 bells, both with and without jump changes? 12345671234561723456127345612374561327456137245613742561374526137456213745612347561324756134275613472561347526134756213475612345761234516723451627345162374516234751623457162345176234512673451263745126347512634571263451726345127634512367451236475123645712364517236451273645123764512346751234657123465172346512734651243765124367512436571243651724365127436512473651246375124635712463517246351274635124763512467351426735146273514672351467325146735216473521674352167345216374521634752163457216345271634521764352176453271645327614532764153276451326745132647513264571326451732645137264531726453712645372164537261453726415372645132764531276453217645231764521376452173654217365241736521473652174365217346521736452176345216735421637542163574216354721635427163542176354216735241637524163572416352741635247163524176352416735214673512465371246531724653127465312476531246753142675314627531467253146752316475321647531264753162475316427531647253164752316745321674531267453162745316724531674253167452316754231675243 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ringing-theory at bellringers.org https://bellringers.org/listinfo/ringing-theory From sh53246 at gmail.com Wed May 22 15:43:35 2019 From: sh53246 at gmail.com (Simon Humphrey) Date: Wed, 22 May 2019 15:43:35 +0100 Subject: [r-t] Superpermutations In-Reply-To: <37808.198.176.82.34.1558515373.squirrel@www.cantab.net> References: <37808.198.176.82.34.1558515373.squirrel@www.cantab.net> Message-ID: <001101d510ac$bcc082d0$36418870$@gmail.com> 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