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Journal Articles Advances in Applied Mathematics Year : 2000

Enumeration of planar constellations

Mireille Bousquet-Melou
Gilles Schaeffer

Abstract

The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let $n\ge 1$, $m \ge 2$, and let $\si_0$ be a permutation of $\Sn_n$ having $d_i$ cycles of length $i$, for $i \ge 1$. We prove that the number of $m$-tuples $(\si_1, \ldots ,\si_m)$ of permutatinos of $\Sn_n$ such that: - $\si_1 \si_2 \cdots \si_m = \si_0$, - the group generated by $\si_1 , \ldots , \si_m$ acts transitively on $\{1, 2, \ldots , n\}$, - $\sum_{i=0}^m c(\si_i) = n(m-1)+2$, where $c(\si_i)$ denotes the number of cycles of $\si_i$, is $$m \ \frac{[(m-1)n-1]!}{[(m-1)n-c(\si_0)+2]!}\ \prod_{i \ge 1} \left[ i {mi-1 \choose i} \right] ^{d_i}.$$ 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 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 {\em transpositions\/} satisfying the above conditions. Indeed, we show that our result implies Hurwitz' theorem. We also briefly discuss its implications for the enumeration of nonequivalent coverings of the sphere.

Dates and versions

inria-00099358 , version 1 (26-09-2006)

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Mireille Bousquet-Melou, Gilles Schaeffer. Enumeration of planar constellations. Advances in Applied Mathematics, 2000, 24 (4), pp.337-368. ⟨10.1006/aama.1999.0673⟩. ⟨inria-00099358⟩
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