Timeline for A bound involving Stirling numbers of the second kind and the asymptotics
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2011 at 0:20 | comment | added | David Moews | \begin{eqnarray*} &\ &\sum_{n\ge 0} \frac{t^n}{(n+r)!} \sum_{k_1+\cdots+k_r=n} \frac{1}{r!} \binom{n+r}{k_1+1 \cdots k_r+1} \prod_{i=1}^r (k_i+1)! x_i^{k_i+1}\\ &= &\frac{1}{r!} \frac{x_1}{1-x_1t}\cdots \frac{x_r}{1-x_rt}. \end{eqnarray*} | |
| Sep 20, 2011 at 0:17 | comment | added | David Moews | From a combinatorial view, this function is a little odd because it mixes $r$-partitions of $\{1,\ldots,n\}$ with $r$-partitions of $\{1,\ldots,n+r\}$. I don't think it will come out to be anything simple. If you changed $S_{n,r}$ to $S_{n+r,r}$, split it up, and pushed it forward through the second sum, you would get something simple: | |
| Sep 17, 2011 at 12:17 | comment | added | Dmitry Kerner |
One related question. I am trying to identify the function whose Taylor series is $\sum_{n\ge0}\frac{S_{n,r}t^n}{(n+r)!}\sum_{\substack{k_1,..,k_r\ge0\\k_1+\cdots+k_r=n}}\prod^r_{i=1}(k_i+1)!x_i^{k_i+1}$. Again, probably this is a simple combinatorics, but I've never learned it. Any ideas about this function? Or some functions with related Taylor series?
|
|
| Sep 16, 2011 at 2:09 | vote | accept | Dmitry Kerner | ||
| Sep 14, 2011 at 6:47 | comment | added | zeb | Doesn't the combinatorial interpretation make it obvious that the limit as r goes to infinity will be 2^n? For large r, the number of ordered partitions where all boxes are of size 1 or 2 will outnumber the number of other partitions by at least a factor of r over a constant depending on n... | |
| Sep 14, 2011 at 5:18 | history | answered | David Moews | CC BY-SA 3.0 |