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