On 14/01/2013 11:22, Joerg Arndt wrote:
I'd like to know whether the following is known (I bet it is, and in a more general context): cf. https://oeis.org/A187761 and https://oeis.org/A179455
Let C(n,x) be the e.g.f of the monotonic-labeled forests on n vertices with rooted trees of height less than n. (A labeled rooted tree is monotonic-labeled if the label of any parent vertex is (strictly) smaller than the label of any offspring vertex.)
I conjecture that the C(n,x) are as follows.
Let C(0,x) = 1 and for n>=1, C(n, x) = exp(integral(C(n-1,x)) )
Well, if F is the e.g.f. for Things Of Size n, then exp F is the e.g.f. for Multisets Of Things Whose Sizes Add Up To n. (The factorials turn into multinomial coefficients.) Which means your conjecture is right. (The integral turns that into "multisets of things whose sizes plus 1 add up to n"; a tree is a forest together with a new node on top.) I'm sure all this is well known to those who know these things well. I am not such a person. -- g