Enumeration of planar constellations
نویسنده
چکیده
The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n > 1, m > 2, and let 0 be a permutation of Sn having di cycles of length i, for i > 1. We prove that the number of m-tuples (1 ; : : : ; m) of permutations of Sn such that: A one-to-one correspondence relates these m-tuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m = 2, we recover a formula of Tutte for the number of Eulerian maps. The proof extends the method applied in 21] to the latter case, and relies on the idea that maps are conjugacy classes of trees. Our result might remind the reader of an old theorem of Hurwitz, giving the number of m-tuples of transpositions satisfying the above conditions. Indeed, we show that our result implies Hurwitz' theorem. We also brieey discuss its implications for the enumeration of nonequivalent coverings of the sphere.
منابع مشابه
A bijection between planar constellations and some colored Lagrangian trees
Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, . . . ) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer [9] who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper...
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تاریخ انتشار 2000