From: Robin Chapman
Subject: Re: Could anyone help me?
Date: Wed, 21 Apr 1999 09:03:33 +1000
Newsgroups: sci.math
Keywords: integrality of binomial-coefficient-like expressions
Dr Acula wrote:
>
> Robin Chapman wrote:
>
> > Olivia Caramello wrote:
> > >
> > > Could anyone help me to prove that
> > >
> > > (n-1)!
> > > -----------
> > > m! (n-m)!
> > >
> > > is a natural number for evey m > > natural numbers)
> > >
> > If n isn't prime, choose a prime factor p and write n = p^r a. Show that
> > if m = p^r then n!/(m!(n-m)!) isn't divisible by p.
> >
> > --
> > Robin Chapman + "Going to the chemist in
> > Department of Mathematics, DICS - Australia can be more
> > Macquarie University + exciting than going to
> > NSW 2109, Australia - a nightclub in Wales."
> > rchapman@mpce.mq.edu.au + Howard Jacobson,
> > http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz
>
> Kummer's theorem and blah blah might be of help in all this, but here
> is another problem :
> prove that
> (2n-1)!
> --------------
> (n-1)! * n!
> is always an integer. This one can do by observing that n and (n-1) are
> coprime.
This is a Catalan number. It equals (2n-2 choose n-1) - (2n-2 choose n).
> But the following stumped me :
> prove that
> (3n)!
> --------------
> n! (n+1)! (n+2)!
> is an integer !!!
> Note that the "sum" in the denominator is 3n+3 whereas in the numerator it is
> 3n.
> I really would like to have a answer to this one, (possibly one that lends
> itself
> to generalization too)
n=1 gives you 6/(1.2.6) = 1/2. This isn't an integer. But *twice* this number
is the number of standard Young tableaux fitting in the 3 by n square.
This is a special case of the hook-length formula for counting standard
Young tableaux. There is also a similar proof to the Catalan numbers above.
--
Robin Chapman + "Going to the chemist in
Department of Mathematics, DICS - Australia can be more
Macquarie University + exciting than going to
NSW 2109, Australia - a nightclub in Wales."
rchapman@mpce.mq.edu.au + Howard Jacobson,
http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz