Timeline for About generalization of stirling numbers of the second kind
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 29, 2017 at 21:49 | comment | added | leonbloy | One reference here pdfs.semanticscholar.org/6f9b/… | |
| Sep 8, 2010 at 16:18 | comment | added | Richard Stanley | The exponential formula (en.wikipedia.org/wiki/Exponential_formula) lets you write down a generating function for the numbers $S_X(k,n)$, the number of partitions of a set of $k$ elements into $n$ classes, where the number of elements of each class belongs to the set $X$ (a subset of the positive integers). Namely $$\sum_{n,k} S_X(k,n)t^n\frac{x^k}{k!} =\exp t\sum_{i\in X}\frac{x^i}{i!}. $$ | |
| Sep 8, 2010 at 11:36 | comment | added | Gjergji Zaimi | You can still write down a recurrence relation $$S(n,k,r)=kS(n-1,k,r)+\binom{n-1}{r-1}S(n-r,k-1,r)$$ by observing where the $n$th term can be inserted. | |
| Sep 8, 2010 at 11:10 | comment | added | Robin Chapman | Of course one can generalize then in this fashion. Whether such a generalization has been studied is a different question. I think they have but don't have a reference to hand. You might extract some sequences from say the $r=2$ example (say for $n=2$, $n=3$ etc.) and search for them in the OEIS. | |
| Sep 8, 2010 at 11:03 | history | asked | Eduardo Lopez | CC BY-SA 2.5 |