A reductive technique for enumerating non-isomorphic planar maps
In [A census of non-isomorphic planar maps, Colloq. Math. Soc. János Bolyai 25, 479-494 (1981; Zbl 0519.05040)] the author reduced the enumeration of planar maps up to orientation-preserving homeomorphisms of the embedding sphere to the enumeration of rooted planar maps done by \textit{W. T. Tutte} [A census of planar maps, Can. J. Math. 15, 249-271 (1963; Zbl 0115.17305)] and found a closed-form formula for the number of such maps with \(n\) edges. \textit{N. C. Wormald} [Counting unrooted planar maps, Discrete Math. 36, 205-225 (1981; Zbl 0467.05034)] and [On the number of planar maps, Can. J. Math. 33, 1-11 (1981; Zbl 0408.57006)] developed an algorithmic method for counting planar maps up to arbitrary homeomorphisms, including orientation-reversing ones. This method yielded no closed-form formula even when restricted to orientation-preserving homeomorphisms. In the article under review, the reductive method of the author [loc. cit.] is generalized to arbitrary homeomorphims by classifying them into five types; the enumeration of planar maps up to arbitrary homeomorphisms is thus reduced to counting rooted planar maps, rooted projective maps and rooted circular maps. A recursive formula for counting rooted projective maps was given in [\textit{E. A. Bender}, \textit{E. R. Canfield} and \textit{R. W. Robinson}, The enumeration of maps on the torus and the projective plane, Can. Math. Bull. 31, No. 3, 251-271 (1988; Zbl 0617.05036)]. In the present paper, rooted circular maps are reduced to certain generalized rooted quadrangular dissections of the disc, and in a final remark a method of enumerating these objects is suggested.
scientific article; zbMATH DE number 932556
scientific article