Talk:Permutohedron

message: 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)
replies: text: 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)
replies: text:  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)
replies: text:  Another, closely related exception: associahedron. Arcfrk 05:19, 22 August 2007 (UTC)
thread_title:  Zonotopes 

message: In math literatute I've usually seen the term spelled "permutAhedron".  Why is it "permutOhedron" here?  Maybe my sample is biased, but is there any particular reason either way?  I'd think, at the least, the opening sentence should have both spellings. Zaslav (talk) 04:46, 25 June 2011 (UTC)
replies: text: I think the "o" spelling is a little more common, and matches the original French spelling, so I it makes sense to keep it as the primary spelling. But I agree that there seems to be enough disagreement over this point to include the other spelling as a valid variant, so I've gone ahead and added it to the lede. —David Eppstein (talk) 05:09, 25 June 2011 (UTC)

text: 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. —David Eppstein (talk) 02:04, 25 February 2017 (UTC)

text: 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.) —David Eppstein (talk) 06:55, 17 August 2022 (UTC)
thread_title:  Spelling 

message: Triangle :
 Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162). - Mitch Harris, Jan 16 2007
I think that this description of the triangle in the OEIS contains two mistakes: It actually is the number of n-k-dimensional faces, and it is the permutohedron of order n (or the n-1-dimensional permutohedron). It's difficult to correct stuff there. Maybe this mistake should be mentioned in references to this sequence. Watchduck (quack) 23:38, 9 August 2013 (UTC)
replies: text: The dimension of this figure is ambiguous. The system of coordinates by which it is defined are n-dimensional, but it lives within an (n − 1)-dimensional hyperplane within the n-dimensional space. —David Eppstein (talk) 19:28, 29 October 2014 (UTC)
replies: text: Changed it to "...of order n (an (n-1)-dimensional polytope)". Watchduck (quack) 22:17, 29 October 2014 (UTC)
replies: text: That sounds good to me. —David Eppstein (talk) 23:13, 29 October 2014 (UTC)
thread_title: OEIS A019538

message: I think since 52.144.34.243 started to  permutohedron and Cayley graph in February 2019, the intro has become more confusing than helpful. I think the  was just fine.
The  that the permutations can be 0-based also makes sense to me. (One could just write "permutations of the first n natural numbers".)
 the old intro is followed by a sentence to the effect, that the elements of the permuted vector can be any rational numbers. Such a generalized permutohedron could be described in the article, if relevant. But I don't think it belongs in the intro.
The statement after that seems to originate from one  by the IP, which was:
The edge-graph of any permutohedron is the Cayley graph of Sn with respect to the generating set of adjacent transpositions (1,2), . . . , (n-1,n).
Zaslav has then   that to:
The edges of a permutohedron (of any kind) are the line segments connecting each pair of permutations that differ only by an adjacent transposition, i.e., the interchange of two adjacent coordinates.
This is clearly a confusion of the permutohedron with the related Cayley graph. At least to me an "adjacent transposition" is clearly one that swaps adjacent places.
But in the permutohedron the edges correspond to all possible transpositions. Just the swapped values are adjacent (i.e. the swapped things on these places).
Here I have written what I hope to be a clarification. (Let me know if you think it's not.)
The intro should be only about the permutohedron. I will change the article accordingly. Watchduck (quack) 20:32, 24 January 2020 (UTC)
replies: text: Thanks. I agree, we should distinguish the permutohedron from the related Cayley graph. —David Eppstein (talk) 06:50, 25 January 2020 (UTC)
replies: text: There is a comment on Commons that adds a new flavor of confusion to this topic: c:Category talk:Permutohedron of order 4 (raytraced) I don't think the claim that the "transposition that swaps the first, second, or final pair changes from vertex to vertex" makes sense, but maybe someone else gets what Adam is trying to say. Watchduck (quack) 11:36, 11 March 2020 (UTC)
thread_title: Expansion of the intro

message: One sentence reads:
"More generally, V. Joseph Bowman (1972) uses that term for any polytope whose vertices have a bijection with the permutations of some set."
So, any polytope with exactly n! vertices would then be called a permutohedron???
Either 1) there is more to what Bowman (1972) proposes, or else 2) this fact is so incompatible with intelligent geometry that it does not belong in the article.216.161.117.162 (talk) 19:10, 26 September 2020 (UTC)
replies: text: You didn't read the source, did you? "Definition: A permutation polyhedron of order n is a convex polyhedron whose extreme points are in one-to-one correspondence with the permutations of order n." He goes on to add a few lines down that he is only actually interested in polyhedra that are connected to optimization problems over permutations. It is used in this article only for the purpose of highlighting the ambiguity of the term "permutation polytope" and is perfectly appropriate in the article for that purpose. —David Eppstein (talk) 19:28, 26 September 2020 (UTC)
thread_title:  This cannot be accurate 

message: Text and explanations                                                                                                                                                                           	Text and explanations
The permutohedron of order n has n! vertices , each of which is adjacent to n − 1 others,                                                                                       	
so the total number of edges is (n − 1)n!/2 .                                                                                                                                   	(Based on what rule?)
Each edge has length , and connects two vertices                                                                                                                                                	
that differ by swapping two coordinates the values of which differ by one.                                                                                                                      	Green edge 123–132 in  swaps 2 and 3.So does 1234–1324 in .
The permutohedron has one facet for each nonempty proper subset S of {1, 2, 3, ..., n},                                                                                                         	Those are all subsets except the empty set and the set itself.The following files show permutohedron-like Cayley graphs.Reading the square matrices by row gives the coordinates.Reading them by column gives the corresponding permutohedron coordinates.In  the facets are the 1-faces a.k.a. edges.In  they are the (2-)faces (hexagons and squares).The facets are represented by two-line matrices. Their first line is the subset.E.g. in the hexagon the top edge corresponds to the subset {2} and the bottom edge to {1, 3}.
consisting of the vertices                                                                                                                                                                      	E.g. the vertices of the hexagons and squares.
in which all coordinates in positions in Sare smaller than all coordinates in positions not in S.                                                                                               	This is indeed difficult. First note that a coordinate is one value, not the tuple.S is the subset corresponding to the facet. Its elements are now called positions.Read the square matrices in this file by column to get the coordinates of the permutohedron.Consider the top left hexagon where S is {2}.So the positions is S are {2} and those not in S are {1, 3, 4}.Look at the six vertices. Their coordinate in position 2 is always 1,which is of course smaller than the other three.Consider the square one the left where S is {2, 3}.So the positions in S are {2, 3} and those not is S are {1, 4}.Look at the four vertices. Their coordinates on positions 2 and 3 are 1 and 2,which is smaller than those on positions 1 and 4, which are 3 and 4.
Thus, the total number of facets is 2n − 2.                                                                                                                                     	The number of subsets is a power of two, and we excluded two.
More generally, the faces of the permutohedron(including the permutohedron itself, but not including the empty set)are in 1-1 correspondence with the strict weak orderings on a set of n items:	has oB(3) = 13 and  has oB(4) = 75 little matrices.
a face of dimension d corresponds to a strict weak orderingin which there are n − d equivalence classes.                                                                        	The equivalence classes (or blocks of the set partition) are the rows of these matrices.
Because of this correspondence, the number of faces is given by the ordered Bell numbers.                                                                                                       	
, , facets, 
    k = 1    2    3    4    5
 n
 1                                     
 2                                    
 3      1                            
 4      1   14                     
 5      1   30  150
The number of (n-k)-dimensional faces in a permutohedron of order nis found in the triangle T(n,k) = k! \cdot S(n,k),where S(n,k) are the Stirling numbers of the second kind                   	
, , facets, 
    k = 1    2    3    4    5
 n
 1                                     
 2                                    
 3      1                            
 4      1   14                     
 5      1   30  150
— shown on the right,together with its row sums, the ordered Bell numbers.                                                                                                                      	
, , facets, 
    k = 1    2    3    4    5
 n
 1                                     
 2                                    
 3      1                            
 4      1   14                     
 5      1   30  150
Did you not understand that just from reading this section? ;) Greetings, Watchduck (quack) 02:14, 5 November 2020 (UTC)
thread_title: ''Vertices, edges, and facets'' in detail

message: truncated octahedron         	truncated cuboctahedron            	truncated icosidodecahedron	source
descriptions in the source   	descriptions in the source         	descriptions in the source 	source
                             	signed p.                          	                           	This term is sometimes used for the convex hull of the signed permutations.e.g. On Flag Vectors...,  Mixed volumes of hypersimplices
p. of type A3                	p. of type B3                      	                           	https://books.google.de/books?id=W_SPdwfPTw8C&pg=PA83#v=onepage&q&f=false
p. for the symmetric group S4	p. for the hyperoctahedral group W3	p. for W(H3)               	https://books.google.de/books?id=Y01d6g5UemQC&pg=PA135#v=onepage&q&f=false
TBC, started by Watchduck (quack) 16:57, 15 November 2020 (UTC)
thread_title: Generalized permutohedron

message: The part about "The permutohedron-like Cayley graph of S4" 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.
Leen Droogendijk (talk) 11:08, 9 June 2021 (UTC)
thread_title: Cayley graph/Permutohedron confusion