Timeline for Permutohedron and triangulation of cube via Eulerian numbers
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 22, 2023 at 1:06 | answer | added | Igor Makhlin | timeline score: 1 | |
| Jun 17, 2023 at 16:08 | vote | accept | Sam Hopkins | ||
| Jun 17, 2023 at 16:07 | answer | added | Richard Stanley | timeline score: 6 | |
| Jun 16, 2023 at 23:47 | comment | added | Richard Stanley | Could something like the following be true? The canonical unimodular triangulation of the $n$-cube is a cone $C(\Delta)$ with apex $(0,0,\dots, 0)$, where $\Delta$ is a triangulation of the facets of the $n$-cube not containing $(0,0,\dots,0)$. Then $\Delta$ is a cone $C(\Gamma)$ with apex $(1,1,\dots,1)$, where $\Gamma$ is a geometric realization of an abstract simplicial complex that is isomorphic to the boundary of the dual permutohedron. | |
| Jun 16, 2023 at 23:46 | comment | added | Tom Copeland | For links among the modified Todd op. and the Eulerian and Pascal polynomials, see Table 2 of "Lattice polytopes, Hecke operators, and the Ehrhart polynomial" by Gunnells & Villegas. This relates Ehrhart polynomials and volumes of simple lattice polytopes, such as the permutahedra. (Ref. in A131758, also see Beck and Robins on the Todd op.) | |
| Jun 16, 2023 at 23:05 | comment | added | Tom Copeland | Interesting question. A090582 contains the f-vectors for the permutahedra and relates them to the Eulerians, to probabilities, and to other geometric models. A123125 relates the Eulerians to 1) the f-and h-polynomials of the stellahedra (A248727, A046802) in turn related to positroids, and 2) the faces in the first barycentric subdivision of the standard n-dimensional simplex (A028246). The Eulerians are connected also via A131758 to core sequences in number theory and analysis (including the polylogarithms). | |
| Jun 16, 2023 at 21:20 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 21:15 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 21:06 | comment | added | Tom Copeland | For the uninitiated: For the Eulerians, see also A123125, A173018. See oeis.org/A019538 for the face polynomials of the permutahedra. As lower triangular matrices $[Prm] = [E][Psc]$, where [Prm] contains the f-polynomial coefficients of the permutahedra; [E], the Eulerians; and [Psc], the binomial coefficients of the Pascal LTM A007318. Umbrally, $E_2(Psc.(x)) =Prm_2(x) = 6 +6x +x^2$. | |
| Jun 16, 2023 at 20:00 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 19:50 | comment | added | Tom Copeland | (Just reminding you of a gripe you made to a recent question of mine. Self-apply or run for office.) As for the unimodular triangulation of the cube? A087127? A290310? | |
| Jun 16, 2023 at 19:43 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 19:37 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 19:31 | comment | added | Sam Hopkins | @TomCopeland: The OEIS entry for the Eulerian numbers is oeis.org/A008292. But the Wikipedia article I linked to is more readable (at least in my opinion). | |
| Jun 16, 2023 at 19:25 | comment | added | Tom Copeland | Just examples and no formulas for the coefficients? No OEIS entries? | |
| Jun 16, 2023 at 16:03 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jun 16, 2023 at 15:57 | history | asked | Sam Hopkins | CC BY-SA 4.0 |