Enumeration of Some Labelled Trees
نویسندگان
چکیده
In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on n] such that the root has exactly k smaller sons. In a rst time we give a bijective proof of the fact that jTn+1;kj = ? n k n n?k. Moreover, we use the family Tn+1;0 of Cayley trees for which the root is smaller than all its sons to give combinatorial explanations of various identities involving n n. We rely this family to the enumeration of minimal factorization of the n-cycle (1; 2; : : : ; n) as a product of transpositions. Finally, we use the fact that jTn+1;0j = n n to prove bijectively that there are 2n n ordered alternating trees on n + 1]. R esum e. Dans cet article nous nous int eressons a l' enum eration d'arbres etiquet es enracin es, en consid erant un nouveau param etre relatif a l'ordre existant entre la racine et ses ls. Soit donc Tn;k la famille des arbres de Cayley sur n] tels que la racine ait exactement k ls qui lui soient inf erieurs. Dans un premier temps, nous donnons une preuve bijective du fait que jTn+1;kj = ? n k n n?k. Ensuite, nous donnons des interpr etations combinatoires de plusieurs identit es relatives a n n en utilisant la famille Tn+1;0 des arbres de Cayley pour lesquels la racine est inf erieure a tous ses ls. Nous lions egalement cette famille a l' enum eration des factorisations minimales transitives du n-cycle (1; 2; : : : ; n) comme produit de transpositions. Finalement, nous utilisons le fait que jTn+1;0j = n n pour d emontrer combinatoirement qu'il y a 2n n arbres ordonn es alternants sur n + 1].
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تاریخ انتشار 2007