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- W1970060470 abstract "The main object of this paper is to give explicit formulas for the number of unlabelled bicolored (unrooted) plane trees with given degree distributions (1 i 1 2 i 2 …; 1 j 1 2 j 2 …). These trees are closely related to the Shabat polynomials, i.e., polynomials over C having at most two critical values (cf. Shabat and Zvonkin, 1994). In the case of planted trees (i.e., rooted at a leaf), this problem was solved by Tutte in 1964, using multivariate Lagrange inversion. Here the key to the solution is the dissymmetry theorem for enriched trees which takes a particularly simple form in the bicolored case and which allows one to get rid of the rooting. We also enumerate those trees in the labelled case, in the unicolored case, as well as when the automorphism group (necessarily cyclic) is of order h equal to (or a multiple of) a given integer k ⩾ 1. L'objectif principal de ce texte est de donner des formules explicites pour le nombre de types d'isomorphie d'arbres plans bicolorés ayant une double distribution (1 i 1 2 i 2 …;1 j 1 2 j 2 …) de degrés donnés à l'avance. Ces arbres sont en étroite relation avec les polynômes de Shabat, c'est-à-dire les polynômes sur C ayant au plus deux valeurs critiques (cf. Shabat and Zvonkin, 1994). Dans le cas des arbres enracinés (pointés en une feuille), ce problème a été résolu par Tutte en 1964 à l'aide de l'inversion de Lagrange multivariée. Ici la clé de la solution réside dans le théorème de dissymétrie pour les arbres enrichis qui, dans le cas bicoloré, prend une forme particulièrement simple et qui permet de se défaire du pointage. Nous dénombrons également ces arbres dans le cas étiqueté, dans le cas unicoloré, ainsi que lorsque le groupe d'automorphismes, nécessairement cyclique, est d'un ordre h égal à (ou multiple de) un entier k ⩾ 1 donné." @default.
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- W1970060470 date "1996-10-01" @default.
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- W1970060470 title "Enumeration of (uni- or bicolored) plane trees according to their degree distribution" @default.
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- W1970060470 doi "https://doi.org/10.1016/s0012-365x(96)83017-2" @default.
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