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{{WikiProject banner shell|class=C|
{{WikiProject Mathematics|importance=low}}
{{WikiProject Polyhedra}}
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{{dyktalk|17 August|2007|entry=...that the six [[permutation]]s of the [[vector]] (1,2,3) form a [[hexagon]] in [[Euclidean space|3d space]], the 24 permutations of (1,2,3,4) form a [[truncated octahedron]] in [[Fourth dimension|four dimensions]], and both are examples of '''[[Permutohedron|permutohedra]]'''?}}
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[[User:LyleRamshaw|LyleRamshaw]] ([[User talk:LyleRamshaw|talk]]) 01:05, 25 February 2017 (UTC)
:I see 827 for A and 759 for O in Google scholar, not enough of a difference to be meaningful. I think scholar is more relevant than general web searches for this topic. And if your Google search found only 29 hits, something is wrong. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 02:04, 25 February 2017 (UTC)
5 years later, and they still seem to be neck and neck. MathSciNet has 101 "permutahedron" hits and 97 "permutohedron" hits. Personally I agree with David, but it's probably just a bias in the literature I'm familiar with. I think it's handled well in any case. [[Special:Contributions/172.250.24.180|172.250.24.180]] ([[User talk:172.250.24.180|talk]]) 04:50, 17 August 2022 (UTC)
:Incidentally, Google scholar also has 2820 hits for "literatute", more than both spellings combined. (Sorry, Z, no intention of criticizing you for the sort of typo we all make, just amused by this instance of [[Muphry's law]].) —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:55, 17 August 2022 (UTC)
==OEIS A019538==
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==New image==
▲}}
For a project at University, I've edited the main file of this article changing the edges' color so that they are not mistaken any more. I'm not sure whether changing it or leaving the old one. Maybe you {{Reply to|Watchduck}} can tell me what to do... [[User:Jordiventura96|Jordiventura96]] ([[User talk:Jordiventura96|talk]]) 00:24, 6 January 2020 (UTC)
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The permutohedron of order {{math|''n''}} has {{math|''n''!}} vertices
The number of edges is {{math|{{sfrac|(''n'' − 1) ''n''!|2}}}}, and their length is [[square root of 2|{{math|{{radic|2}}}}]].
Two connected vertices differ by swapping two coordinates
(In [[:File:Symmetric group 4; permutohedron 3D; transpositions (1-based).png|the example image]] the vertices {{math|(3, 2, 1, 4)}} and {{math|(2, 3, 1, 4)}}
The number of [[Facet (mathematics)|facets]] is {{math|2<sup>''n''</sup> − 2}}, because they correspond to non-empty proper [[subset]]s {{math|''S''}} of {{math|{{mset|1 … ''n''}}}}.
The vertices of a facet corresponding to subset {{math|''S''}} have in common, that their coordinates on places in {{math|''S''}} are
{| class="wikitable collapsible collapsed"
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|colspan="5"|<div style="width: 1250px;">
For order 3 the facets are the 6 edges, and for order 4 they are the 14 <abbr title="(here in the usual sense of 2-faces)">faces</abbr>. <br>
The coordinates on places {{math|''i''}} where {{math|''i'' ∈ ''S''}} are marked with a dark background. It can be seen, that they are
The square on top corresponds to the subset {{math|{{mset|
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!colspan="2" style="border-right: 3px solid #aaa;"| order 3
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! 3-element subsets
|- style="text-align: center;"
| [[File:Permutohedron 3 subsets 1 (first).svg|250px]]
|style="border-right: 3px solid #aaa;"| [[File:Permutohedron 3 subsets 2 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 1 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 2 (first).svg|250px]]
| [[File:Permutohedron 4 subsets 3 (first).svg|250px]]
|}
More generally, the [[k-face|faces]] of dimensions 0 (vertices) to {{math|''n'' − 1}} (the permutohedron itself) correspond to the [[Weak ordering#Strict weak orderings|strict weak orderings]] of the set {{math|{{mset|1 … ''n''}}}}. So the number of all faces is the {{math|''n''}}-th [[ordered Bell number]].<!--<ref name="zface">See, e.g., {{harvtxt|Ziegler|1995}}, p. 18.</ref>-->
A face of dimension {{math|''d''}} corresponds to an ordering with {{math|''k'' {{=}} ''n'' − ''d''}} equivalence classes.
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The vertex labels {{nowrap|a{{hsp}}{{!}}{{hsp}}b{{hsp}}{{!}}{{hsp}}c{{hsp}}{{!}}{{hsp}}d}} interpreted as permutations {{nowrap|(a,{{hsp}}b,{{hsp}}c,{{hsp}}d)}} are those forming the Cayley graph.
The before mentioned facets are among these faces
The first one is used as the subset {{math|''S''}} assigned to the facet, so the ordering is {{math|(''S'', ''S''<sup>''c''</sup>)}}.
The images below show how the facets of the {{math|''n''}}-permutohedron correspond to the [[simplex|simplical]] projection of the {{math|''n''}}-[[hypercube|cube]].<br>
The binary coordinate labels correspond to the subsets {{math|''S''}}, e.g.
</div>
|- style="text-align: center;"
| [[File:Triangular cube shadow with bit patterns (first).svg|250px]]
| [[File:Tetrahedral tesseract shadow with bit patterns (first).svg|300px]]
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<math>T(n,k) = k! \cdot \left\{{n\atop k}\right\}</math> with <math>\textstyle \left\{{n\atop k}\right\}</math> representing the [[Stirling numbers of the second kind]]<br>
It is shown on the right together with its row sums, the [[ordered Bell number]]s.
:Why are you starting your rewrite with zero sources? All content should be based on reliable sources. That means, find sources first, then write according to what the sources say about the subject. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 02:21, 11 November 2020 (UTC)▼
'''Added source:''' {{citation| last = Lancia | first = Giuseppe | title = Compact extended linear programming models | publisher = Springer | location = Cham, Switzerland | year = 2018 | isbn = 978-3-319-63975-8}}
▲:Why are you starting your rewrite with zero sources? All content should be based on reliable sources. That means, find sources first, then write according to what the sources say about the subject. The article in its current state is not unsourced. Any rewrite should not be a step backwards in that respect. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 02:21, 11 November 2020 (UTC)
::I just left them out here, because I don't like stray references on talk pages. Anyway, I just added them. <small>(And removed them again after [https://en.wikipedia.org/w/index.php?title=Permutohedron&diff=prev&oldid=990049897 adding the change].)</small>
::The statement that was criticized as unreadable ("consisting of the vertices in which all coordinates...") does not currently have a source. I found one here: https://www.mimuw.edu.pl/~jarekw/pdf/Permutahedron.pdf (p. 105) There are two ways to assign the subsets to the facets. I used the second block of the <abbr title="strict weak ordering">SWO</abbr>, but this source uses the first block, and so does the current text in the article. Guess I will have to adapt to that. (Done.) --[[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 16:13, 15 November 2020 (UTC)
==Generalized permutohedron==
{| class="wikitable" style="text-align: center;"
! [[File:Polyhedron great rhombi 4-4 max.png|70px]]<br>[[truncated octahedron]]
! [[File:Polyhedron great rhombi 6-8 max.png|70px]]<br>[[truncated cuboctahedron]]
! [[File:Polyhedron great rhombi 12-20 max.png|70px]]<br>[[truncated icosidodecahedron]]
!rowspan="2"| source
|-
!colspan="3"| descriptions in the source
|-
|
| signed p.
|
| This term is sometimes used for the convex hull of the [[signed permutation]]s.<br><small>e.g. [https://link.springer.com/article/10.1007/PL00009428 ''On Flag Vectors...''], [https://arxiv.org/abs/1406.6440 ''Mixed volumes of hypersimplices'']</small>
|-
| p. of type A<sub>3</sub>
| p. of type B<sub>3</sub>
|
| https://books.google.de/books?id=W_SPdwfPTw8C&pg=PA83#v=onepage&q&f=false
|-
| p. for the [[symmetric group]] S<sub>4</sub>
| p. for the [[hyperoctahedral group]] W<sub>3</sub>
| p. for [[Icosahedral symmetry|W(H<sub>3</sub>)]]
| https://books.google.de/books?id=Y01d6g5UemQC&pg=PA135#v=onepage&q&f=false
|}
TBC, started by [[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 16:57, 15 November 2020 (UTC)
==Cayley graph/Permutohedron confusion==
The part about "The permutohedron-like Cayley graph of S<sub>4</sub>" and a large part of the page https://commons.wikimedia.org/wiki/Category:Permutohedron_of_order_4_(raytraced)#Permutohedron_vs._Cayley_graph is superfluous.
One only needs to realize that applying a permutation on one side swaps places, and applying it on the other side swaps values.
Both compared graphs are Cayley graphs, one is left-Cayley, the other is right-Cayley.
[[User:Leen Droogendijk|Leen Droogendijk]] ([[User talk:Leen Droogendijk|talk]]) 11:08, 9 June 2021 (UTC)
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