Timeline for Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2021 at 16:14 | comment | added | Wolfgang | @TimothyBudd I see you were really lucky as well! The only numbers yielding this OEIS entry are in this column. And all the others don't yield anything useful in OEIS. | |
| Dec 30, 2021 at 15:36 | answer | added | Wolfgang | timeline score: 1 | |
| Dec 30, 2021 at 13:14 | comment | added | Timothy Budd | @MartinRubey I put the number 12869120 in OEIS, returning exactly one result: A137668 related to the coefficients of $\tan(x \arctan(t))$. After noting several more coincidences I worked my way to the $\frac{1}{2}\sinh(2 y\operatorname{tanh} x)$. | |
| Dec 30, 2021 at 12:30 | comment | added | Martin Rubey | @TimothyBudd I'd be extremely interested in how you came up with your guess! | |
| Dec 29, 2021 at 9:50 | comment | added | Martin Rubey | Actually, I suspect that the generating function should be $\frac{1}{2}\left(1+\exp(y\log\frac{1+x}{1-x})\right)$, because of the empty permutation. But to decide, a definition would be necessary. | |
| Dec 28, 2021 at 14:58 | vote | accept | Wolfgang | ||
| Dec 28, 2021 at 14:29 | comment | added | Martin Rubey | It should be possible to interpret this generating function combinatorially, by rewriting it as $\frac{1}{2}\exp\left(y\log\frac{1+x}{1-x}\right)$. The term $\log\frac{1+x}{1-x}$ is the generating function for cycles of odd length, either blue or red, so in total we get 'permutations' with all cycles of odd length, and each cycle coloured blue or red, up to swapping colours. Each cycle has weight $y$. | |
| Dec 28, 2021 at 13:59 | comment | added | Martin Rubey | @TimothyBudd, and the even part is $\frac{1}{2}\cosh(2y\,\mathrm{arctanh} x)$, so in total we obtain $\frac{1}{2}\exp(2y\,\mathrm{arctanh} x)$. | |
| Dec 28, 2021 at 11:47 | comment | added | Timothy Budd | The bivariate generating function seems to be given by $\sum_{n,m\geq 0} \frac{a_{n,m}}{(2n+1)!} x^{2n+1}y^{2m+1} = \frac{1}{2} \sinh(2 y \operatorname{arctanh} x)$. Or was this the origin of your numbers? | |
| Dec 28, 2021 at 8:31 | comment | added | Wolfgang | @Bma the numbers are somewhat related to the integrals $\int\limits_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$, see here mathoverflow.net/questions/271526/…. | |
| Dec 28, 2021 at 8:18 | comment | added | Wolfgang | @LSpice The numbers came up when evaluating numerically certain integrals and then doing quite a bit of manipulations with those values. Of course with the assumption that there is some nice closed form. | |
| Dec 28, 2021 at 0:03 | comment | added | LSpice | To further @Bma's question, presumably there is a definition of this array that allows you to produce these numbers, but that happens not to be of the form you want. What is the definition? | |
| Dec 27, 2021 at 23:41 | answer | added | Martin Rubey | timeline score: 9 | |
| Dec 27, 2021 at 21:41 | comment | added | Bma | Where did this originally come up? | |
| Dec 27, 2021 at 21:30 | history | asked | Wolfgang | CC BY-SA 4.0 |