33 research outputs found
Condorcet domains of tiling type
A Condorcet domain (CD) is a collection of linear orders on a set of
candidates satisfying the following property: for any choice of preferences of
voters from this collection, a simple majority rule does not yield cycles. We
propose a method of constructing "large" CDs by use of rhombus tiling diagrams
and explain that this method unifies several constructions of CDs known
earlier. Finally, we show that three conjectures on the maximal sizes of those
CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic
Pl\"ucker environments, wiring and tiling diagrams, and weakly separated set-systems
For the ordered set of elements, we consider the class \Bscr_n of
bases of tropical Pl\"ucker functions on such that can be
obtained by a series of mutations (flips) from the basis formed by the
intervals in . We show that these bases are representable by special
wiring diagrams and by certain arrangements generalizing rhombus tilings on the
-zonogon. Based on the generalized tiling representation, we then prove that
each weakly separated set-system in having maximum possible size
belongs to \Bscr_n, thus answering affirmatively a conjecture due to Leclerc
and Zelevinsky. We also prove an analogous result for a hyper-simplex
.Comment: 47 pages. In this revision we add an Appendix containing results on
weakly separated set-systems in a hyper-simplex and related subject
Planar flows and quadratic relations over semirings
Adapting Lindstr\"om's well-known construction, we consider a wide class of
functions which are generated by flows in a planar acyclic directed graph whose
vertices (or edges) take weights in an arbitrary commutative semiring. We give
a combinatorial description for the set of "universal" quadratic relations
valid for such functions. Their specializations to particular semirings involve
plenty of known quadratic relations for minors of matrices (e.g., Pl\"ucker
relations) and the tropical counterparts of such relations. Also some
applications and related topics are discussed.Comment: 35 pages. This is the revised version accepted for publication in J.
Algebraic Comb. (The final publication is available at springerlink.com.)
Also in the Appendix we add the assertion that the function of minors of any
matrix over a field is generated (using Lindstr\"om method) by flows in a
weighted planar acyclic directed grap
Higher Bruhat orders of types B and C
We propose versions of higher Bruhat orders for types and . This is
based on a theory of higher Bruhat orders of type~A and their geometric
interpretations (due to Manin--Shekhtman, Voevodskii--Kapranov, and Ziegler),
and on our study of the so-called symmetric cubillages of cyclic zonotopes.Comment: 22 pages, 12 figures, a reference is added, a misprint in commutative
diagram is correcte
On wiring and tiling diagrams related to bases of tropical Plücker functions
We consider the class of bases of tropical Plücker functions on the Boolean -cube such that can be obtained by a series of flips from the basis formed by the intervals of the ordered set of elements. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on a zonogon