Permutations and Combinations
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Permutations and Combinations
by Friday's Child-2
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Hi,
I'm working on something and remember how nice and friendly you guys out here used to be. My brain is getting old. Maybe one of you will remember about stuff like this and help me out. Does anyone know or recollect a function or formula that sums over all the possible permutations available to a given set of objects? Take, say, the group of 6 objects: A, B, C, D, E, F. I already know that you can order them in the various chains A-B-C-D-E-F, B-C-A-D-E-F and so on and so forth and I already know that this total is 6! or 6 x 5 x 4 x 3 x 2 x 1 or 720. I also know that if you want all the triples, then it's 6P3 which is 6!/(6-3)! which is 6!/3! which is 6 x 5 x 4 so giving 120 chains such as B-D-F, C-E-B and so forth. So that now makes 120 plus 720 possibilities for this group. When you've gone through every single possibility and collected up all singles, doubles, triples, quartets, quintuples and sextets, you get 1,237 total possible groupings. What I cannot remember is if there is a simple and elegant way to compute ALL possible permutations in that way for any arbitrary group of size n. I want to represent and compute the totality of nPr as r varies from 1 to n for any n. This kind of thing plays a part in error-checking, in probability theory and in all sorts of great stuff like that, so I'm pretty sure that one of you out there will know -- if anyone is still listening these days. It's just that I've forgotten way too many things to work it out from first principles these days ... and nor can I find a reference that tells me this. If someone can indicate a way to find out, I would be obliged. And ... do please keep it simple. My days are filled with feeding bunny rabbits and other such high-level stuff these days!! Thanks. With every good wish to one and all, and I hope things are good with all of you, (K)ofi (B)usia ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Friday's Child-2
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Hi Wout,
It only helps partly. But it assures me that I am on the right path. We need two things: (a) an algorithm for determining the sum of all the factorials for any r; and (b) an algorithm for determining the sum of all the combinations at that same r. Or to put it another way, what that first person who spoke up is effectively saying is that we need to compute the number of orderings at each given r as r ranges from 1 to n; and that we then need to add to that the number of combinations and we will have the number of permutations. But that is not really saying anything because the formula for the number of combinations is already known to be [n!/(n-r)! x r!]. Even I can remember that much!! An algorithm certainly exists for finding a factorial, because every spreadsheet can do this. So one would think that an algorithm also exists for finding the sum of all the factorials. But I haven't found it yet. As far as I can see, the solution is surely that the total number of permutations at any n is: n + [n x (n - 1)] + [n x (n - 1) x (n - 2)] + [n x (n - 1) x (n - 2) x (n - 3)] + ........ through all the r's as n declines until (n - r) = 2; and then we stop and add 1 to our result. Which is why I was thinking last night that the binomial theorem comes into it. But I really just need to express the sum of such a series elegantly as r ranges from n down to 2. I've had some sleep now. I can only hope that my mind is a bit fresher. So I should be able to figure it out today once I've fed my daughter's rabbit and other all those other such essential life functions that befall me these days. I used to be able to solve such problems with such ease, but they just don't seem to come up much these days, and I've lost the basic skills. Thanks for the reference, though, it told me that I wasn't going nuts!! I hope your son is bringing you all the joy and pleasure that my own children have brought to me. Have more, if you can. That's what rabbits do. With every good wish, (K)ofi (B)usia Wm Blommers wrote: > Hi Kofi, > Could this be of any help? > http://bytes.com/forum/thread514194.html > Wout > >> Datum: 06/07/08 05:35 AM >> Van: "Friday's Child" >> Aan: "Nord Modular mailing list" >> CC: >> Onderwerp : [NM] Permutations and Combinations >> >> Hi, >> >> I'm working on something and remember how nice and friendly you guys out >> here used to be. My brain is getting old. Maybe one of you will remember >> about stuff like this and help me out. >> >> Does anyone know or recollect a function or formula that sums over all >> the possible permutations available to a given set of objects? >> >> Take, say, the group of 6 objects: A, B, C, D, E, F. >> I already know that you can order them in the various chains >> A-B-C-D-E-F, B-C-A-D-E-F and so on and so forth and I already know that >> this total is 6! or 6 x 5 x 4 x 3 x 2 x 1 or 720. >> >> I also know that if you want all the triples, then it's 6P3 which is >> 6!/(6-3)! which is 6!/3! which is 6 x 5 x 4 so giving 120 chains such as >> B-D-F, C-E-B and so forth. >> So that now makes 120 plus 720 possibilities for this group. >> >> When you've gone through every single possibility and collected up all >> singles, doubles, triples, quartets, quintuples and sextets, you get >> 1,237 total possible groupings. >> >> What I cannot remember is if there is a simple and elegant way to >> compute ALL possible permutations in that way for any arbitrary group of >> size n. >> I want to represent and compute the totality of nPr as r varies from 1 >> to n for any n. >> >> This kind of thing plays a part in error-checking, in probability theory >> and in all sorts of great stuff like that, so I'm pretty sure that one >> of you out there will know -- if anyone is still listening these days. >> It's just that I've forgotten way too many things to work it out from >> first principles these days ... and nor can I find a reference that >> tells me this. If someone can indicate a way to find out, I would be >> obliged. And ... do please keep it simple. My days are filled with >> feeding bunny rabbits and other such high-level stuff these days!! >> >> Thanks. >> >> With every good wish to one and all, and I hope things are good with all >> of you, >> (K)ofi (B)usia >> ___________________________________________________________ >> Nord-modular mailing list >> Nord-modular@... >> http://mail.electro-music.com/mailman/listinfo/nord-modular >> >> Members: use http://mail.electro-music.com/mailman/options/nord-modular >> to change subscription status, digest options, password, topics, etc. >> >> Patches sent to the Nord-Modular mailing list may not be redistributed >> without the express consent of the author/creator. >> ___________________________________________________________ >> >> >> >> ------------------------------------------------------------------------ >> >> ___________________________________________________________ >> Nord-modular mailing list >> Nord-modular@... >> http://mail.electro-music.com/mailman/listinfo/nord-modular >> >> Members: use http://mail.electro-music.com/mailman/options/nord-modular >> to change subscription status, digest options, password, topics, etc. >> >> Patches sent to the Nord-Modular mailing list may not be redistributed >> without the express consent of the author/creator. >> ___________________________________________________________ >> ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Jurgen Post
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Hello everybody,
first of all, its interesting that some people are still reading this forum and there are certainly some who need these kind of mathematics for what ever and furthermore, who understand this stuff. BUT, what has this to do with the Nord Modular ? I am sure that if you wrote this kind of text in any other forum, there would be flames without end. I am quite new to this forum, because I got my Modular just a few months ago and was interested, what other people think and do with their machine. But what I can read here (whenever there is something to read) is a bit disappointing, although I admit, the Modular is not really a brand new synth and those who use it mostly have no more interest in searching and trying. Just my 2 cents. Wishing you happiness Juergen -----Ursprungliche Nachricht----- Von: nord-modular-bounces@... [mailto:nord-modular-bounces@...]Im Auftrag von Friday's Child Gesendet: Sonntag, 6. Juli 2008 14:35 An: blommoo@...; Nord Modular mailing list Betreff: Re: [NM] Permutations and Combinations Hi Wout, It only helps partly. But it assures me that I am on the right path. ... ... ... ... ... ... ... ... ad infinitum ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Friday's Child-2
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Hi Jurgen,
Jurgen Post wrote: > first of all, its interesting that some people are still reading > this forum Amazing, isn't it?!! > and there are certainly some who need these kind of > mathematics for what ever Yup. > and furthermore, who understand this > stuff. > I don't, always, but I like to try to. > BUT, what has this to do with the Nord Modular ? > Synthesizing in the end reduces to computing in an effort to emulate the sounds we hear, and the requires the production of algorithms for all kinds of purposes. Playing around with filters, and designing them, and emulating various kinds of synths requires exercises in integration, differentiation, differencing, exponentiation and all such other things. There are lots of patches exploring integration and differentiation and so forth. This is also a routing problem in error detection and so forth. But, in another way, if you wanted to make a path that selected from instruments and or volumes and or other patches at random you could have say a four-way switch to route things. How many different kinds of combinations of sounds and or volumes and or notes and whatever else could you work through using every combination of addition and so forth from one or two or three four-way switches? I like to work with percussion and cross-rhythms myself, and started on this because I turned on my NM after a long period away, and was trying to make a patch that worked its way through a rhythm sequence of 2 beats, 3 beats. 4 beats and 5 beats, and I started wondering how many different kinds of cycles I could create and it just grew from that. I don't know if there are patches exploring factorials, but I'm intrigued. There's also a kind of entropy in the permutations you choose to discard as the numbers get larger. I'm curious to explore what effects that has on sounds and combinations. I'm a bit insane that way. Then the computer crashed and I realized that I hadn't switched on the Modular for so long I'd forgotten that it was essential to save all the time. But, I'll do it again. > I am sure that if you wrote this kind of text in any other forum, > there would be flames without end. > Yes. But then that was the nice thing about the old NM list. You could ask questions like that, and eventually someone would make a patch with a thought they had on a stupid question that someone had asked. > I am quite new to this forum, Welcome. > because I got my Modular just a few > months ago and was interested, The NM1 was a really great synth. If you have doubts about the NM2, do just send yours over and I'll happily test it out for you and give you a report. I am very thorough, so you would need to let me have it for a while. About 3 years should cover it. If that sounds attractive to you -- to be parted from your NM2 for three years so I can "test" it for you and tell you if it's worth having one, let me know and I'll send you my address and you can ship it over. I'll probably end up telling you that while it might have been worth your while spending the money on it, I'm not sure it's good enough for you to keep, and that I probably need yet another three-year "testing" period to complete my evaluations. I really am very very thorough. It's worth it, however. I recommend my own services very highly. > what other people think and do with > their machine. > A lot of great things were done with old NM1. > But what I can read here (whenever there is something to read) Sadly, very little on this mailing list these days. > is > a bit disappointing, Agreed. > although I admit, the Modular is not really a > brand new synth No, it isn't. What makes it new are things you can think of to do with it. Like I started working on above. > and those who use it mostly have no more interest > in searching and trying. > What do you mean by that? The people I remember on this list were ALWAYS searching and trying. > Wishing you happiness > Same to you. Stay well now. With every good wish, (K)ofi (B)usia ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Grant Ransom
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This mailing list was always extremely open-minded about it's discussions,
which is what made it *so* good. From obsessive pedantry, right through the spectrum to trial-and-error experimentalism, everyone appreciated the value in everyone elses perspectives. So, some of the discussions will be (however few, nowadays) extremely abstract to a newer user, but the custom here is/was to ask for explanations. Grant. On Sun, 06 Jul 2008 16:09:10 +0200, Jurgen Post <juergen-post@...> wrote: > Hello everybody, > > first of all, its interesting that some people are still reading > this forum and there are certainly some who need these kind of > mathematics for what ever and furthermore, who understand this > stuff. > > BUT, what has this to do with the Nord Modular ? > > I am sure that if you wrote this kind of text in any other forum, > there would be flames without end. > > I am quite new to this forum, because I got my Modular just a few > months ago and was interested, what other people think and do with > their machine. > > But what I can read here (whenever there is something to read) is > a bit disappointing, although I admit, the Modular is not really a > brand new synth and those who use it mostly have no more interest > in searching and trying. > > Just my 2 cents. > > Wishing you happiness > > Juergen > > > > -----Ursprungliche Nachricht----- > Von: nord-modular-bounces@... > [mailto:nord-modular-bounces@...]Im Auftrag von Friday's > Child > Gesendet: Sonntag, 6. Juli 2008 14:35 > An: blommoo@...; Nord Modular mailing list > Betreff: Re: [NM] Permutations and Combinations > > > Hi Wout, > > It only helps partly. But it assures me that I am on the right path. > ... > ... > ... > ... > ... > ... > ... > ... > ad infinitum > ___________________________________________________________ > Nord-modular mailing list > Nord-modular@... > http://mail.electro-music.com/mailman/listinfo/nord-modular > > Members: use http://mail.electro-music.com/mailman/options/nord-modular > to change subscription status, digest options, password, topics, etc. > > Patches sent to the Nord-Modular mailing list may not be redistributed > without the express consent of the author/creator. > ___________________________________________________________ ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Grant Ransom
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I forgot to say: Welcome! Grant. ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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Re: Permutations and Combinations
by Friday's Child-2
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Friday's Child wrote:
> As far as I can see, the solution is surely that the total number of > permutations at any n is: > n + [n x (n - 1)] + [n x (n - 1) x (n - 2)] + [n x (n - 1) x (n - 2) x > (n - 3)] + ........ through all the r's as n declines until (n - r) = 2; > and then we stop and add 1 to our result. > Nope. Now I think it's: n! + n + [n x (n - 1)] + [n x (n - 1) x (n - 2)] + [n x (n - 1) x (n - 2) x (n - 3)] + ........ + and so declining until (n - r) = 2. So the total for 6 should be 1,956, I think. With every good wish, (K)ofi (B)usia ___________________________________________________________ Nord-modular mailing list Nord-modular@... http://mail.electro-music.com/mailman/listinfo/nord-modular Members: use http://mail.electro-music.com/mailman/options/nord-modular to change subscription status, digest options, password, topics, etc. Patches sent to the Nord-Modular mailing list may not be redistributed without the express consent of the author/creator. ___________________________________________________________ |
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