A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
Combinatorial Algorithms: For Computers and Hard Calculators
Combinatorial Algorithms: For Computers and Hard Calculators
Generating functions for generating trees
Discrete Mathematics
Walks confined in a quadrant are not always D-finite
Theoretical Computer Science - Random generation of combinatorial objects and bijective combinatorics
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Partially directed paths in a wedge
Journal of Combinatorial Theory Series A
Classifying lattice walks restricted to the quarter plane
Journal of Combinatorial Theory Series A
Analytic Combinatorics
Two non-holonomic lattice walks in the quarter plane
Theoretical Computer Science
The enumeration of prudent polygons by area and its unusual asymptotics
Journal of Combinatorial Theory Series A
Weakly directed self-avoiding walks
Journal of Combinatorial Theory Series A
Some New Self-avoiding Walk and Polygon Models
Fundamenta Informaticae - Lattice Path Combinatorics and Applications
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A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Prea in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them. However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well known to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even D-finite. The fourth class-general prudent walks-is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-D-finite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed.