-----Original Message----- Behalf Of R. William Gosper Sent: Wednesday, February 04, 2004 2:57 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] limit question
Apropos Catalan numbers, a few weeks ago I conjectured the integerhood of
2 binomial(n, i) (n + 2 i + 3)! ------------------------------- . (i + 1)! (i + 2)! (n + 3)!
This is the special case C(3,i,n-i) of
k /===\ (n + k)! (n + m)! | | (j - 1)! (n + j m + m)! C(m, k, n) := ----------------- | | ----------------------- , n! (n + m + k)! | | (m + j - 1)! (n + j m)! j = 1 which I also conjecture integral.
This is true. Consider the following partition p(m,k,n)=(n+m,m,...,m) of n+m*(k+1), where m is repeating k times. It is easy to see that C(m,k,n) equals the dimension of the irreducible representation of S_(n+m*(k+1)) corresponding to p(m,k,n) calculated using hook length formula. Another formula for C(m,k,n) is m - 1 /====\ (n + m k + m)! | | i! -------------- | | ------------------------ n! | | (k + i)! (n + k + i + 1) | | i = 0 Alec Mihailovs http://webpages.shepherd.edu/amihailo/