618 research outputs found
A two-sided analogue of the Coxeter complex
For any Coxeter system of rank , we introduce an abstract boolean
complex (simplicial poset) of dimension that contains the Coxeter
complex as a relative subcomplex. Faces are indexed by triples , where
and are subsets of the set of simple generators, and is a
minimal length representative for the parabolic double coset . There
is exactly one maximal face for each element of the group . The complex is
shellable and thin, which implies the complex is a sphere for the finite
Coxeter groups. In this case, a natural refinement of the -polynomial is
given by the "two-sided" -Eulerian polynomial, i.e., the generating function
for the joint distribution of left and right descents in .Comment: 26 pages, several large tables and figure
On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution
to a convex optimization problem. We show that if Sigma(R, C) is sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) with high probability is close to a
particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among
all matrices with row sums R, column sums C and entries between 0 and 1.
Similar results are obtained for 0-1 matrices with prescribed row and column
sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
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