A simple grammar.
For n > 0, Sum_{k=1..n} a(k)*Sum_{i=0..n-k} (-1)^i*k^i*Stirling1(n-i,k)/(i!*(n-i)!) = delta(n,1). -
Vladimir Kruchinin, Feb 08 2012
Number of tail-trees of weight n. A tail is a pairing of a block of a set partition p with an element of some other block. A tail-tree on p is composed of a root block r, a tail-tree on each block of a set partition of the remaining blocks, and a tail from each of their roots to r.
On any set partition of weight n and length m, the total number of tail-forests with k components is equal to binomial(m-1, k-1)*n^(m-k). (End)
Number of nonempty forests of rooted labeled hypertrees with a total number of vertices equal to n.
E.g., n=1: Only one forest is possible, which is {1}, the forest with one hypertree with one vertex.
n=2: Three forests are possible: {1,2}, the forest with two hypertrees, each having one vertex labeled 1 for one on the hypertree and 2 for the other hypertree; the forest {1-2}, with only one hypertree, with two vertices rooted at 1; and the forest {2-1}, with only one hypertree, with two vertices rooted at 2. (End)
