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Aryabhatta
Uberpuzzler      Gender: Posts: 1321 Reciprocal Product Summation   « on: Jul 13th, 2009, 1:53am » Quote Modify

Let N be given positive integer. Find the sum:

Sigma  ((a+1) (b+1) (c+1))-1

where the sum ranges over the set of solutions of

a + b + c = N, a, b, c being non-negative integers. IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler      Gender: Posts: 1948 Re: Reciprocal Product Summation   « Reply #1 on: Jul 13th, 2009, 12:06pm » Quote Modify

 hidden: This is the coefficient of xn+3 in ( xa+1/(a+1))^3 = [-log(1-x)]3, ...which is listed in sloane as 6|s(n+3,3)|/(n+3)!. Discovering this result may be one thing, but proving it isn't so hard. Let c(n,k) be the number of permutations of an n-element set with k cycles. Inductively, suppose c(n,k)k! xn/n! = [-log(1-x)]k, which is cleary true for k=0.  Then c(n,k)k! xn/n!) xi/i  = xn c(n-i,k)k!/((n-i)! * i),   so we must show that   c(n,k+1)(k+1)! = [n(n-1)...(n-i-1)/i]*c(n-i,k)k!   The left side counts the number of ways to permute n elements with (k+1) cycles, and then order the cycles.  But there are n(n-1)...(n-i-1)/i ways the first cycle can have length i, and then c(n-i,k)k! ways to pick the remaining k cycles.

Note the inverse results:
 hidden: s(n,k)k! xn/n! = [log(1+x)]k S(n,k)k! xn/n! = [exp(x)-1]k
 « Last Edit: Jul 13th, 2009, 12:07pm by Eigenray » IP Logged
Aryabhatta
Uberpuzzler      Gender: Posts: 1321 Re: Reciprocal Product Summation   « Reply #2 on: Jul 13th, 2009, 12:41pm » Quote Modify

That is correct. I found it interesting that the generating functions for powes of ln (1-x) give us stirling numbers! IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler      Gender: Posts: 1948 Re: Reciprocal Product Summation   « Reply #3 on: Sep 13th, 2009, 5:30pm » Quote Modify

Just came across a nicer proof in van Lint & Wilson:

1/k! [ log(1+x) ]k is the coefficient of zk in

exp( z log(1+x) ) = (1+x)z
= C(z,n) xn
= (z)n/n! xn
= xn/n! s(n,k)zk
= zk s(n,k) xn/n! IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler      Gender: Posts: 1948 Re: Reciprocal Product Summation   « Reply #4 on: Sep 14th, 2009, 5:30pm » Quote Modify

Just realized that in the context of Joyal species these are both one-liners:

c(n,k) counts the number of ways to break an n-set into k (non-empty) cycles, so c(n,k) xn/n!  =  C(x)k/k!,
where C(x) = (n-1)! xn/n! = -log(1-x), since there are (n-1)! ways make an n-set into a single cycle.

S(n,k) counts the number of ways to break an n-set into k non-empty sets, so S(n,k) xn/n! = F(x)k/k!,
where F(x) = n>0  xn/n! = ex-1, since there is 1 way to make an n-set into a non-empty set, for n > 0. IP Logged

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