Planar Maps as Labeled Mobiles
Copyright policy )Planar Maps as Labeled Mobiles
We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.
Distance in graphs, Statistical Mechanics (cond-mat.stat-mech), Secondary 05A15, 05C30 (Primary) 05A15, 05C05, 05C12, 68R05 (Secondary), Exact enumeration problems, generating functions, Eulerian planar maps, FOS: Physical sciences, Primary 05C30, Enumeration in graph theory, labeled trees, Trees, enumeration, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], FOS: Mathematics, Mathematics - Combinatorics, quadrangulations, Combinatorics (math.CO), Schaeffer's bijection, Primary 05C30; Secondary 05A15, 05C30 (Primary) 05A15, 05C05, 05C12, 68R05 (Secondary), Condensed Matter - Statistical Mechanics
Distance in graphs, Statistical Mechanics (cond-mat.stat-mech), Secondary 05A15, 05C30 (Primary) 05A15, 05C05, 05C12, 68R05 (Secondary), Exact enumeration problems, generating functions, Eulerian planar maps, FOS: Physical sciences, Primary 05C30, Enumeration in graph theory, labeled trees, Trees, enumeration, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], FOS: Mathematics, Mathematics - Combinatorics, quadrangulations, Combinatorics (math.CO), Schaeffer's bijection, Primary 05C30; Secondary 05A15, 05C30 (Primary) 05A15, 05C05, 05C12, 68R05 (Secondary), Condensed Matter - Statistical Mechanics
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