Timeline for What is the formula for $\mathcal P_{n}^{k} (a_{1}, a_{2}, ...)$, defined by Peter Luschny?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6 at 15:45 | comment | added | Math123 | @PeterTaylor O, I see. Thank you very much! | |
| Jul 6 at 15:31 | comment | added | Peter Taylor | @Math123, no: Ferrers diagrams are a subset which is strict for $n \ge 3$. But in the French convention they would be stalagmites instead of stalactites. | |
| Jul 6 at 15:30 | history | edited | Peter Taylor | CC BY-SA 4.0 |
Add diagram example
|
| Jul 6 at 9:19 | vote | accept | Math123 | ||
| Jul 6 at 9:19 | comment | added | Math123 | @PeterTaylor Thank you very much! So these stalactite diagrams are English convention of Ferrers diagrams (for example as here: edwardmpearce.github.io/tutorial-partitions/intro/visualization/…)? | |
| Jul 5 at 15:39 | history | edited | Peter Taylor | CC BY-SA 4.0 |
added 869 characters in body
|
| Jul 5 at 10:17 | comment | added | Peter Taylor | $A_{n,k}$ is expressed as a sum over partitions of $n$ into $k$ parts, whereas $P_n^k$ is expressed as a sum over partitions of $n$ with largest part $k$. It's possible that there's a connection through conjugation of partitions. I suggest thinking about how to express them through labellings of Ferrers diagrams, but I don't promise that that will be useful. | |
| Jul 5 at 8:57 | comment | added | Math123 | P.S. If you think that should be a new separated question, please let me know. | |
| Jul 5 at 8:57 | comment | added | Math123 | Thank you very much. Only one question: I recently discovered this article (arxiv.org/abs/2203.02868) on ArXiv where DeMoivre polynomials $A_{n, k}$ are defined (equation 1.2 and definiton 1.1). They satisfy very similar connections to Stirling numbers of both kinds (equations 2.21 and 2.22). The formula involving Stirling cycle numbers is the same, the formula involving Stirling set numbers is a bi different because of the factorials. So my question is how are $A_{n, k}(a_{1}, ...)$ and $P_{n}^{k}(a_{1}, ...)$ connected? | |
| Jul 5 at 7:18 | history | answered | Peter Taylor | CC BY-SA 4.0 |