Timeline for Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 22 at 0:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 17 at 19:21 | comment | added | Tom Copeland | Re the third involution, see tree diagrams on p. 9 of "Tree hook length formulae, Feynman rules and B-series by Jones and Yeats (arxiv.org/abs/1412.6053), oeis.org/A355201, and $[A^{(-1)} ]$ in my extended answer below. | |
| Jul 24 at 23:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 26 at 22:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 27, 2023 at 21:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 3, 2023 at 0:12 | comment | added | Tom Copeland | An example of this pair of Laurent series are at the top of p. 24 of "Löwner equations and dispersionless hierarchies" by Takashi Takebe, Lee-Peng Teo, and Anton Zabrodin (arxiv.org/abs/math/0605161). See also eqn. 4.1 on p. 8. | |
| Jul 31, 2023 at 17:34 | comment | added | Tom Copeland | @SamHopkins, not directly. I've seen the paper before. It seems to deal with only purely triangular Riordan matrices--not an infinite set of indeterminates and partition polynomials of the sort I illustrate. Some OEIS reductions/specializations below reference a more directly related paper by Paul Barry. Btw, you might already know this, but the $[A^{(m}]$ and $[N]^m$ form a realization of an infinite dihedral grp, which is interesting since finite dihedral groups are used in the Coxeter group formalism to characterize the associahedra and noncrossing partitions. I posted an MSE-Q on this. | |
| Jul 30, 2023 at 20:07 | comment | added | Sam Hopkins | I don't know if it is directly related to your question, but you might be interested in the "pseudo-involutions in the Riordan group": see, e.g., arxiv.org/abs/2112.11595 | |
| Jul 30, 2023 at 20:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 1, 2023 at 20:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 23, 2023 at 18:23 | comment | added | Tom Copeland | Adjusting the overall sign of a polynomial, the involution III is eqn 1.3 on p. 70 of "Lagrange Inversion and Schur Functions" by Lenart, who references "Symmetric Functions and Hall Polynomials" by Macdonald (p. 35, Ex. 24). | |
| Feb 25, 2023 at 18:04 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Elaborated on definitions
|
| Feb 25, 2023 at 11:54 | comment | added | YCor | Starting the post with a definition of "involution" might be useful. | |
| Feb 25, 2023 at 2:05 | answer | added | Tom Copeland | timeline score: 0 | |
| Jun 26, 2022 at 16:11 | comment | added | Tom Copeland | Example III--essentially Schur's self-convolution expansion coefficients $-b_(n+1)=\frac{K_{n+1,n}}{n}$, or $\frac{K_{n+1}^(n)}{n}$--to be characterized in A355201. | |
| Jun 26, 2022 at 4:09 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Dependency on a_0 corrected.
|
| Jun 11, 2022 at 20:03 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Expanded set of indeterminates for last example, changed a tag
|
| Jun 11, 2022 at 3:55 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added new involution
|
| May 17, 2022 at 17:32 | comment | added | Tom Copeland | @SamHopkins: The relations among the LIF, the LPT, and compositional polynomials such as the Bell partition polynomials, OEIS A036040, are given in my pdf "Lagrange a la Lah" (2011). The partition polynomials that are inverse to the Bell are noted in the OEIS entry. | |
| May 14, 2022 at 20:50 | comment | added | Tom Copeland | @SamHopkins: Certainly related to the Bell polynomials and similar compositional partition polynomials, but the CPPs don't provide an involution since $f(g(x))$ is not typically the same as $g(f(x))$. | |
| May 14, 2022 at 20:43 | comment | added | Tom Copeland | @SamHopkins, the o.g.f. version of example I, A263633 as noted in the example, covers the elementary and complete homogeneous functions (see my related comment in that OEIS entry.) | |
| May 14, 2022 at 20:41 | comment | added | Sam Hopkins | Sorry, maybe that is exactly the same as your example #1. | |
| May 14, 2022 at 19:54 | comment | added | Sam Hopkins | Maybe the involution $\omega$ on the ring $\Lambda$ of symmetric functions which swaps the elementary $e_n$ and complete homogeneous $h_n$ symmetric functions would give another natural example. | |
| May 14, 2022 at 17:43 | history | asked | Tom Copeland | CC BY-SA 4.0 |