26,408 research outputs found
Dyck paths and pattern-avoiding matchings
How many matchings on the vertex set V={1,2,...,2n} avoid a given
configuration of three edges? Chen, Deng and Du have shown that the number of
matchings that avoid three nesting edges is equal to the number of matchings
avoiding three pairwise crossing edges. In this paper, we consider other
forbidden configurations of size three. We present a bijection between
matchings avoiding three crossing edges and matchings avoiding an edge nested
below two crossing edges. This bijection uses non-crossing pairs of Dyck paths
of length 2n as an intermediate step.
Apart from that, we give a bijection that maps matchings avoiding two nested
edges crossed by a third edge onto the matchings avoiding all configurations
from an infinite family, which contains the configuration consisting of three
crossing edges. We use this bijection to show that for matchings of size n>3,
it is easier to avoid three crossing edges than to avoid two nested edges
crossed by a third edge.
In this updated version of this paper, we add new references to papers that
have obtained analogous results in a different context.Comment: 18 pages, 4 figures, important references adde
Stable marriages and search frictions
Stable matchings are the primary solution concept for two-sided matching markets with nontransferable utility. We investigate the strategic foundations of stability in a decentralized matching market. Towards this end, we embed the standard marriage markets in a search model with random meetings. We study the limit of steady-state equilibria as exogenous frictions vanish. The main result is that convergence of equilibrium matchings to stable matchings is guaranteed if and only if there is a unique stable matching in the underlying marriage market. Whenever there are multiple stable matchings, sequences of equilibrium matchings converging to unstable, inefficient matchings can be constructed. Thus, vanishing frictions do not guarantee the stability and efficiency of decentralized marriage markets
A bijection between the set of nesting-similarity classes and L & P matchings
Matchings are frequently used to model RNA secondary structures; however, not
all matchings can be realized as RNA motifs. One class of matchings, called the
L P matchings, is the most restrictive model for RNA secondary structures
in the Largest Hairpin Family (LHF). The L P matchings were enumerated in
by Jefferson, and they are equinumerous with the set of
nesting-similarity classes of matchings, enumerated by Klazar. We provide a
bijection between these two sets. This bijection preserves noncrossing
matchings, and preserves the sequence obtained reading left to right of whether
an edge begins or ends at that vertex.Comment: 9 pages, 7 figure
Applications of Graphical Condensation for Enumerating Matchings and Tilings
A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text
revised, but content essentially the sam
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