Analytic models and ambiguity of context-free languages
Theoretical Computer Science
On chromatic and dichromatic sum equations
Discrete Mathematics
GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable
ACM Transactions on Mathematical Software (TOMS)
The Number of Degree-Restricted Rooted Maps on the Sphere
SIAM Journal on Discrete Mathematics
Basic analytic combinatorics of directed lattice paths
Theoretical Computer Science
Generating functions for generating trees
Discrete Mathematics
Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling
ACM Transactions on Algorithms (TALG)
Analytic Combinatorics
Asymptotic enumeration of constellations and related families of maps on orientable surfaces
Combinatorics, Probability and Computing
Two non-holonomic lattice walks in the quarter plane
Theoretical Computer Science
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We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial @g"M(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q0,4 is of the form 2+2cos(j@p/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial P"M(q,@n), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two ''catalytic'' variables. To our knowledge, this is the first time such equations are being solved since Tutte@?s remarkable solution of properly q-colored triangulations.