FW: Re: Sequence : 2-dimensional faces of a polytope

[Yuval Dekel]

>From: Victor Reiner <reiner at math.umn.edu>
>To: Yuval Dekel <dekelyuval at hotmail.com>
>Subject: Re: Sequence : 2-dimensional faces of a polytope Date: Sun, 11 Jan 
>2004 00:15:51 -0600 (CST)
>Dear Yuval,
>
>   I don't know if you've already had a reply from millions
>of others, but I was gone for a week and just saw your mail...
>
>The entire facial structure of the Birkhoff polytope is pretty
>well-known, and described, for example, in old papers by Richard
>Brualdi (for which I can't look up the reference at home right
>now).   It is also reviewed, I believe, in the book by Lovasz
>and Plummer called "Matching theory".  The basic idea is this:
>
>   A face is indexed by an edge-subgraph G of the complete bipartite
>   graph K_{n,n} for which the edge set E(G) can be written is a
>   union of perfect matchings.
>
>   Containment of faces corresponds to _reverse_ containment
>   of the edges sets E(G).
>
>   The dimension of a face has to do with the the number of connected
>   components of the graph G, but I've forgotten the formula (and
>   it is certainly in Brualdi's papers, probably in Lovasz and Plummer).
>
>Once one has this dimension formula for a face, describing the vertices,
>edges, two-dimensional faces is pretty easy, I think.
>
>Best wishes,
>Vic
>
>___________________________________________________________________
>Vic Reiner                      (612) 625-6682 (office, voice mail)
>School of Mathematics           (612) 626-2017 (math dept. FAX)
>127 Vincent Hall
>Univ. of Minnesota
>Minneapolis, MN 55455
>
>Office: Vincent Hall 256
>Web page: http://www.math.umn.edu/~reiner
>
>
>On Tue, 6 Jan 2004, Yuval Dekel wrote:
>
> > Apart from the SeqFan mailing list this is off topic but let me ask :
> >
> > Sequence A059760 in the OEIS :
> > http://www.research.att.com/projects/OEIS?Anum=A059760
> >
> > gives a description of the edges (one-dimensional faces) in the convex
> > polytope of real n X n doubly stochastic matrices.
> >
> > Can someone give a description of the 2-dimensional faces of this 
>polytope
> > and their number ?
> >
> > TIA,
> > Yuval
> >
> > _________________________________________________________________

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