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

[Yuval Dekel]

Perhaps someone in the list has access to the papers of Brualdi and can 
compute the number of
2-dimensional faces .

TIA,
Yuval


>From: Richard Brualdi <brualdi at math.wisc.edu>
>Reply-To: brualdi at math.wisc.edu
>To: dekelyuval at hotmail.com (Yuval Dekel)
>Subject: Re: Sequence : 2-dimensional faces of a polytope
>Date: Thu, 19 Feb 2004 14:09:33 -0600 (CST)

>I have never counted the number of such faces, but the faces can only be
>rectangles and triangles, and the description of such faces in terms of 
>their
>"representing matrices" is given in the papers I wrote with Gibson some 
>years
>ago (see below). Using that description it should be easy to count them.
>
>Hope this helps.
>
>Richard Brualdi
>
>
>     \item The convex polyhedron of doubly stochastic matrices:  I.
>Applications of the Permanent Function, (with
>     P. Gibson).  J. Combinatorial Theory (A) 22 (1977), pp. 194-230.
>
>     \item The convex polyhedron of doubly stochastic matrices:  II. The
>graph of   (with P. Gibson).  J.
>     Combinatorial Theory (B) 22 (1977), pp. 175-198.
>
>     \item The convex polyhedron of doubly stochastic matrices: III.
>Affine and combinatorial properties of  n
>     (with P. Gibson).  J. Combinatorial Theory (A) 22 (1977), pp.
>338-351.
>
>     \item Convex polyhedra of doubly stochastic matrices:  IV. (with P.
>Gibson). Linear Algebra and Its
>     Applications 15 (1976), pp. 15-172.
>
>     \item The assignment polytope (with P. Gibson).  Mathematical
>Programming II (1976), pp. 97-101.
>
>According to Yuval Dekel:
> >
> > Dear professor Brualdi ,
> > I asked the following qeustion in the seqfan mailing list :
> >
> > 
>----------------------------------------------------------------------------------------------------------------------------------------
> > 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
> >
> > 
>-----------------------------------------------------------------------------------------------------------------------------------------
> > Let me ask you if there is a "nice" formula for the number of 
>2-dimensional
> > faces of the Birkhoff polytope similiar to the formulas for the number 
>of
> > vertices and edges .
> >
> > TIA,
> > Yuval Dekel
> >


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>* Richard A. Brualdi,                                               *
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