34 research outputs found
Dyck paths with coloured ascents
We introduce a notion of Dyck paths with coloured ascents. For several ways
of colouring, we establish bijections between sets of such paths and other
combinatorial structures, such as non-crossing trees, dissections of a convex
polygon, etc. In some cases enumeration gives new expression for sequences
enumerating these structures.Comment: 14 pages, 11 figure
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Suballowable sequences of permutations
We discuss a notion of "allowable sequence of permutations" and show a few combinatorial results and geometric applications
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
Disjoint compatibility graph of non-crossing matchings of points in convex position
Let be a set of labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
. Two such matchings, and , are disjoint compatible if they do
not have common edges, and no edge of crosses an edge of . Denote by
the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each , the connected components of
form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 figure
On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution
International audienc