Abstract
We consider unicellular maps, or polygon gluings, of fixed genus. In FPSAC '09 the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the ``recursive part'' of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, or the Lehman-Walsh/Goupil-Schaeffer formulas. Thanks to previous work of the second author this also leads us to a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Nous considèrons des cartes orientèes à une face de genre fixé. à SFCA'09 le premier auteur a introduit une bijection rècursive envoyant une carte unicellulaire vers un arbre, ce qui permet d'obtenir des formules ènumèratives pour les cartes à une face (et en particulier la prèsence des nombres de Catalan). Dans l'article ici prèsent, et en nous appuyant sur la bijection ci-dessus, nous obtenons une incarnation très simple des cartes à une face comme des paires formèes d'un arbre plan et d'une permutation d'un certain type. Toutes les formules prècèdemment connues dècoulent aisèment de cette nouvelle incarnation, donnant des preuves bijectives dans un cadre unifié. Pour certaines de ces formules, telles que la rècurrence de Harer-Zagier ou les formules de Lehman-Walsh/Goupil-Schaeffer, nous obtenons la première preuve bijective connue. Par ailleurs, en combinant notre approche avec des travaux du second auteur, nous obtenons une nouvelle expression pour les polynômes de Stanley qui donnent certaines èvaluations des caractères du groupe symètrique.
Highlights
A unicellular map is a connected graph embedded in a surface in such a way that the complement of the graph is a topological disk
Guillaume Chapuy and Valentin Feray and Eric Fusy these connections have turned the enumeration of unicellular maps into an important research field
The goal of this paper is to present a new bijection between unicellular maps and surprisingly simple objects which we call C-decorated trees
Summary
A unicellular map is a connected graph embedded in a surface in such a way that the complement of the graph is a topological disk. The goal of this paper is to present a new bijection between unicellular maps and surprisingly simple objects which we call C-decorated trees (these are merely plane trees equipped with a certain kind of permutation on their vertices) This bijection is based on the previous work of the first author [3]: we explicitly describe the “recursive part” appearing in this work. C-decorated trees are so simple combinatorial objects that all formulas follow from our bijection as an immediate corollary or easy exercise Another interesting application of this bijection is a new explicit way of computing the so-called Stanley character polynomials, which are nothing but the evaluation of irreducible characters of the symmetric groups, properly normalized and parametrized. We do not obtain a “closed form” expression (there is no reason to believe that such a form exists!), we express Stanley character polynomials as the result of a term-substitution in free cumulants, which are another meaningful quantity in representation theory of symmetric groups
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
