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Generating functions of embedded trees and lattice paths. (English) Zbl 1244.05060
Summary: J. Bouttier, P. Di Francesco and E. Guitter [Nucl. Phys., B 663, No. 3, 535–567 (2003; Zbl 1022.05022)] introduced a method for solving certain classes of algebraic recurrence relations arising the context of maps and embedded trees. The aim of this note is to apply their method, consisting of a suitable ansatz and (computer assisted) guessing, to three problems, all related to the enumeration of lattice paths. First, we derive the generating function of a family of embedded binary trees, unifying some earlier results in the literature. Second, we show that several enumeration problems concerning so-called simple families of lattice paths can be solved without using the kernel method. Third, we use their method to (re-)derive the length generating function of three vicious walkers and osculating walkers.
MSC:
05C05 Trees
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
05C30 Enumeration in graph theory
05C12 Distance in graphs
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