Kuo, Eric H. Applications of graphical condensation for enumerating matchings and tilings. (English) Zbl 1043.05099 Theor. Comput. Sci. 319, No. 1-3, 29-57 (2004). Summary: 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 planar bipartite graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions. Cited in 10 ReviewsCited in 88 Documents MSC: 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05B45 Combinatorial aspects of tessellation and tiling problems 05A17 Combinatorial aspects of partitions of integers 05A15 Exact enumeration problems, generating functions Keywords:Tilings; Perfect matchings; Plane partitions; Generating functions × Cite Format Result Cite Review PDF Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: a(n) = 2^(n*(n-1)/2). a(n) = Fibonacci(n)*Fibonacci(n+2). References: [1] D. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Mathematical Association of America, Washington, DC, 1999, pp. 197-199.; D. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Mathematical Association of America, Washington, DC, 1999, pp. 197-199. · Zbl 0944.05001 [2] Carlitz, L., Rectangular arrays and plane partitions, Acta Arith., 13, 29-47 (1967) · Zbl 0168.01502 [3] Cohn, H.; Elkies, N.; Propp, J., Local statistics for random domino tilings of the Aztec diamond, Duke Math. J., 85, 117-166 (1996) · Zbl 0866.60018 [4] Elkies, N.; Kuperberg, G.; Larsen, M.; Propp, J., Alternating-sign matrices and domino tilings (Part I), J. Algebr. Combin., 1, 111-132 (1992) · Zbl 0779.05009 [5] Gessel, I.; Viennot, G., Binomial determinants, paths, and hook length formulae, Adv. in Math., 58, 3, 300-321 (1985) · Zbl 0579.05004 [6] Grensing, D.; Carlsen, I.; Zapp, H.-Chr., Some exact results for the dimer problem on plane lattices with non-standard boundaries, Philos. Mag. A, 41, 777-781 (1980) [7] MacMahon, P., Memoir on the theory of partitions of numbers—Part V. Partitions in two-dimension space, Philos. Trans. Roy. Soc. London, 211, 75-110 (1912), Reprinted in Percy Alexander MacMahon: Collected Papers, George E. Andrews (Ed.), Vol. 1, MIT Press, Cambridge, MA, 1978, pp. 1328-1363 · JFM 42.0236.21 [8] Proctor, R., Odd symplectic groups, Invent. Math., 92, 307-332 (1988) · Zbl 0621.22009 [9] J. Propp, Enumerations of matchings: problems and progress, New Perspectives in Geometric Combinatorics, Vol. 38, MSRI Publications, Cambridge University Press, Cambridge, UK, 1999, pp. 255-291.; J. Propp, Enumerations of matchings: problems and progress, New Perspectives in Geometric Combinatorics, Vol. 38, MSRI Publications, Cambridge University Press, Cambridge, UK, 1999, pp. 255-291. · Zbl 0937.05065 [10] Propp, J., Generalized domino-shuffling, Theoret. Comput. Sci., 303, 267-301 (2003) · Zbl 1052.68095 [11] Werman, M.; Zeilberger, D., A bijective proof of Cassini’s identity, Discrete Math., 58, 109 (1986) · Zbl 0578.05004 [12] B.-Y. Yang, Three enumeration problems concerning Aztec diamonds, Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1991.; B.-Y. Yang, Three enumeration problems concerning Aztec diamonds, Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1991. [13] Zeilberger, D., Reverend Charles to the aid of Major Percy and Fields-medalist Enrico, Amer. Math. Monthly, 103, 501-502 (1996) · Zbl 0856.15007 [14] Zeilberger, D., Dodgson’s determinant-evaluation rule proved by two-timing men and women, Electron. J. Combin., 4, 2, R22 (1997) · Zbl 0886.05002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.