2 formatting triangle

Hi.

$$a_{n,k} = \sum_{1 = m_1 < m_2 < \cdots < m_k = n}\ \prod_{j=2}^k S(m_j, m_{j-1})$$.

where the $S(n, k)$ are Stirling numbers of the second kind. The triangle begins:

1,
0, 1,
0, 1, 3,
0, 1, 13, 18,
0, 1, 50, 205, 180,
...

Something very much like this is listed on oeis.org as "Triangle of coefficients from fractional iteration of $e^x - 1$" (and it is doing such fractional iteration that piques my interest in this sequence, and I got the formula while fiddling around with the one mentioned in my earlier question here: http://mathoverflow.net/questions/64450/is-there-a-theory-about-these-kinds-of-recurrence-equations-is-this-a-known-form). The two look to be identical except for possible indexing differences and the presence/absence of leading zeroes, but I don't have a proof (yet?).

Now, my questions are: does my $a_{n,k}$ have a combinatorial interpretation of some sort, and if so, what is it? Is there a "simpler" formula for it (i.e. one that does not involving summing over up to an exponential or worse amount of terms, even if it is not symbolically shorter)? And finally, is there a combinatorial interpretation of a product of Stirling numbers $S(m_2, m_1) S(m_3, m_2) \cdots S(m_k, m_{k-1})$ where $m_1 < m_2 < \cdots < m_k$, and if so, what is it?

1

# Is there a combinatorial interpretation of this triangle sequence? Is there a "simpler" formula?

Hi.

$$a_{n,k} = \sum_{1 = m_1 < m_2 < \cdots < m_k = n}\ \prod_{j=2}^k S(m_j, m_{j-1})$$.
where the $S(n, k)$ are Stirling numbers of the second kind. The triangle begins:
Something very much like this is listed on oeis.org as "Triangle of coefficients from fractional iteration of $e^x - 1$" (and it is doing such fractional iteration that piques my interest in this sequence, and I got the formula while fiddling around with the one mentioned in my earlier question here: http://mathoverflow.net/questions/64450/is-there-a-theory-about-these-kinds-of-recurrence-equations-is-this-a-known-form). The two look to be identical except for possible indexing differences and the presence/absence of leading zeroes, but I don't have a proof (yet?).
Now, my questions are: does my $a_{n,k}$ have a combinatorial interpretation of some sort, and if so, what is it? Is there a "simpler" formula for it (i.e. one that does not involving summing over up to an exponential or worse amount of terms, even if it is not symbolically shorter)? And finally, is there a combinatorial interpretation of a product of Stirling numbers $S(m_2, m_1) S(m_3, m_2) \cdots S(m_k, m_{k-1})$ where $m_1 < m_2 < \cdots < m_k$, and if so, what is it?