53 research outputs found
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
The sorting order on a Coxeter group
Let be an arbitrary Coxeter system. For each word in the
generators we define a partial order--called the {\sf -sorting
order}--on the set of group elements that occur as
subwords of . We show that the -sorting order is a
supersolvable join-distributive lattice and that it is strictly between the
weak and Bruhat orders on the group. Moreover, the -sorting order is a
"maximal lattice" in the sense that the addition of any collection of Bruhat
covers results in a nonlattice. Along the way we define a class of structures
called {\sf supersolvable antimatroids} and we show that these are equivalent
to the class of supersolvable join-distributive lattices.Comment: 34 pages, 7 figures. Final version, to appear in Journal of
Combinatorial Theory Series
The core of games on ordered structures and graphs
In cooperative games, the core is the most popular solution concept, and its
properties are well known. In the classical setting of cooperative games, it is
generally assumed that all coalitions can form, i.e., they are all feasible. In
many situations, this assumption is too strong and one has to deal with some
unfeasible coalitions. Defining a game on a subcollection of the power set of
the set of players has many implications on the mathematical structure of the
core, depending on the precise structure of the subcollection of feasible
coalitions. Many authors have contributed to this topic, and we give a unified
view of these different results
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