simple recursion help

[Thorsten Kampe]

* Hung Jung Lu (2004-11-15 07:05 +0100)
> Thorsten Kampe <thorsten at thorstenkampe.de> wrote:
>>> (1) "Variation" is the same as "permutation".
>> 
>> Sorry, no.
> 
> Sorry, yes. Please learn to accept the fact that a word
> ("permutation", in this case) can have several definitions. You are
> not the Pope of mathematics, and there is none. Different people
> define it different ways. Your definition is by no way the only
> accepted definition. You have been raised one school of
> notation/terminology, other people have been raised in another school
> of notation/terminology. What the French call "body" ("corps"), the
> American call it "field" ("champ") as in "the Real Number Field, the
> Complex Number Field". Many examples like that.
> 
>> * variations   without repetition = P(n, k)
> 
> Funny, "variation" starts with the letter "v", where do you think the
> "P" as in your "P(n, k)" come from? Surely not from "Pariation",
> right? The fact that you see the "P(n, k)" notation shows you that
> many people call this function "permutation". You simply were raised
> in a particular school of terminology and were not aware that another
> school of terminology existed.
> 
> What you have called ""variation with repetition", other people call
> it "string". As I said, you are not the Pope of mathematics, and don't
> expect other people will agree with your terminology.
> 
> Learn to accept the fact that what you call "variation", other people
> call it "permutation". Like it or not, it's a fact. Now, re-read the
> following sentence:
> 
>>> (1) "Variation" is the same as "permutation".
> 
> and try to understand what I was saying:
> 
>     Your "variation" == Other people's "permutation"
> 
> is that clear, now?
> 
> Please check
> http://mathworld.wolfram.com/BallPicking.html
> which I have pointed out in my earlier message and which you did not
> bother to read, at all. I would say the majority of students in the
> U.S.A. are trained with the terminology convention I use. Surely the
> usage of the term "variation" is also popular, but I would say that at
> present it constitutes the minority, not the majority.

Reading some articles in the Wikipedia I have to partly (well, maybe
even mostly) agree with you. Variations are in fact other peoples'
permutations in /popular/ math.

But I think the fact you're missing (and I tried to explain) is that
this use is deprecated and abandoned in scientific math for more than
a hundred years (since the rise of modern set theory).

Combinatorics (in its old use) is much older than set theory. It's
still one of the most applied fields (in its "ball picking" sense) for
gambling and other things. That's why the old habits die so slowly.
I've searched books from the fifties where variations still were
called "unordered combinations".

But combinatorics isn't "ball picking with and without 'putting back'"
anymore and so the old use of permutation isn't "official" anymore.

Two reasons for that are immediately understandable:

1. The old use of permutation always meant "unordered k-sets without
repetition". There simply was no name for "variations with
repetition".

That's why you stated 'For people that use the term "variation", the
term "permutation" is reserved for the special case V(n, n). Neither
name is right for the original question". And that's why people had to
invent names like "string" for this "thing without a name" - while in
scientific combinatorics it already had a name: variation with
repetition (or cartesian product in set theory).

2. Combinatorics isn't just "ball picking" anymore. It includes
"modern style" permutations (n!) - with and without repetition. That's
why the old use of permutation (that was used in "ball picking") isn't
used in scientific math and combinatorics anymore. There simply was no
way to teach combinatorics using "old" and "new" permutations.

And one word at the end: I'm not the pope of math. I did study math
for two years and even if I had a degree this wouldn't make me the
pope. I was trying to explain things, not to declare them.

Before I wrote my "cvp" program I did an exhaustive research (not on
the Internet) because I was very confused about the popular
terminology.

If you use (and think in) the new terminology combinatorics and it's
several cases are very clear and distinct: permutations, variations
and combinations with and without repetition. Six clearly categorized
cases.

And remember: the "new" use isn't that new (a hundred years and more).
If you continue to use the old terms of popular math you're
manifesting the existing confusion (and make people invent new names
like "string").

Thorsten