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Enumeration of events on plane lattices. (English. Russian original) Zbl 0956.05008

Dokl. Math. 58, No. 2, 257-259 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 362, No. 5, 595-597 (1998).
From the text: The problem of enumeration of events that occur on an integer lattice whose edges are binary variables that interact through a common node is considered. The interaction is defined by the list of admissible binary configurations formed by the edges emanating from a node, and the problem of enumeration consists in the representation of the number of occurring events (the realizations of the lattice) as a function of the dimensions and the topology of the lattice and the type of interaction between edges. On a one-dimensional lattice, the problem reduces to the enumeration of realizations of the corresponding Markov chain and is easily solved by applying an appropriate matrix method. For two-dimensional lattices, the problem becomes substantially more complicated, and its solution, as a rule, is closely related to a specific model of interaction. The matrix method developed below that essentially employs the matrix description of the two-dimensional-lattice topology seems to be more universal.
We analyze lattices of three modifications: a rectangular lattice on a plane, a rectangular diagonally oriented lattice on a cylinder, and a lattice on a cylinder with the honeycomb topology.

MSC:

05A15 Exact enumeration problems, generating functions
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