Talk:Permutohedron

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A fact from Permutohedron appeared on Wikipedia's Main Page in the Did you know? column on 17 August 2007.

[edit] Zonotopes

The article claims that every permutohedron is a zonotope. This is morally true, but unfortunately a zonotope is defined to be a 3-dimensional thing, while a permutohedron can be n-dimensional for any n. This can't be fixed in the permutohedron article, it needs to be fixed in the zonotope article. Adam1729 01:16, 17 August 2007 (UTC)

Actually, I think that's a misreading of the zonotope article (a redirect to zonohedron): it defines a zonohedron as a three-dimensional thing, and a zonotope as arbitrary dimensional. But the zonotope definition was buried in the middle of the article; I just made it more prominent. —David Eppstein 04:08, 18 August 2007 (UTC)

Thanks. The confusion is partly because a ...hedron is 3d whereas a ...tope is n-d, except for a permutohedron. Maybe we should say this. Adam1729 01:12, 19 August 2007 (UTC)

Another, closely related exception: associahedron. Arcfrk 05:19, 22 August 2007 (UTC)

[edit] Omnitruncated n-simplices

By induction I believe this is true, even if no sources to support it:

The permutohedron of order n is an omnitruncated (n − 1)-simplex.:

n	Uniform polytope (Omnitruncated (n-1)-simplex) Schläfli symbol group: Coxeter-Dynkin diagram	Picture	Tessellation A~n-1 or Pn Coxeter group	Vertices n!	Facets 2n-2	Facet counts by type
2	Interval t0{} A1:		Apeirogon	2	2
3	Hexagon (Truncated triangle) t0,1{3} A2:		Hexagonal tiling	6	6	2*3 {}
4	Truncated octahedron (Omnitruncated tetrahedron) t0,1,2{3,3} A3:		Bitruncated cubic honeycomb	24	14	2*4 t0,1{3} + 6 {}x{}
5	Omnitruncated 5-cell t0,1,2,3{3,3,3} A4:			120	30	2*5 t0,1,2{3,3} + 2*10 t0,1{3}x{}
6	Omnitruncated 5-simplex t0,1,2,3,4{3,3,3,3} A5:			720	62	2*6 t0,1,2,3{3,3,3} + 2*15 t0,1,2{3,3} + 20 t0,1{3}xt0,1{3}
7	Omnitruncated 6-simplex t0,1,2,3,4,5{3,3,3,3,3} A6:			5040	126	2*7 t0,1,2,3,4{3,3,3,3} + 2*21 t0,1,2,3{3,3,3} + 35 t0,1,2{3,3}xt0,1{3}
...
n	Omnitruncated (n-1)-simplex t0,1,..,n-2{3n-2} An-1:...			n!	2n-2	Σ[i=0..n-3] C(n,i)t0,...,i-1{3i-1}xt0,...,n-2-i{3n-2-i}

Tom Ruen (talk) 00:13, 25 November 2007 (UTC). (Expanded into a table) Tom Ruen (talk) 07:24, 10 December 2007 (UTC)