Re: a question of counting

From: Ed Beroset (beroset_at_mindspring.com)
Date: 03/16/05 

Date: Wed, 16 Mar 2005 00:06:40 GMT

David Einstein wrote:
> Ed Beroset wrote:
>
>> Very interesting. I'm also interested in knowing if there is a
>> generating function for the set of numbers which is representable (or
>> conversely for the set of numbers which is not.)
>
>
> Well, the generating function which counts the number of ways that a
> number is representable is very simple f(x)=1/((1-x^7)(1-x^11)(1-x^16))
> in your case. Kevin Woods' technique says that the generating function
> which merely indicates whether or not a number is representable is
> f(x)=(1-x^32-x^44-x^49+x^60+x^65)/((1-x^7)(1-x^11)(1-x^16)) (or
> something like that, I may have horked the arithmetic), and something
> similar is true in general, though it does get considerably more
> complicated sometimes for more than 3 coins. This result never ceases
> to amaze me.

That IS amazing. I am gaining a new appreciation for just how much math
I don't know yet. :-)

> If all this interests you, Matthias Beck and Sinai Robbins are writing a
> text aimed at undergraduates (it will be a Springer UTM book) and was
> available on Matt's web site last time I checked, but math.sfsu.edu is
> not talking to me this morning, so I cannot give a precise URL.

Got it, thanks! They have the first 8 chapters finished and on-line at
Beck's home page. For those following along, the URL is
http://math.sfsu.edu/beck/index.html

Thanks again!

Ed