Mathematics – Combinatorics
Scientific paper
2003-04-07
Theoretical Computer Science, Vol. 319/1-3 (2004), pp. 29-57
Mathematics
Combinatorics
25 pages; 21 figures Corrected typos; Updated references; Some text revised, but content essentially the same
Scientific paper
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.
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