Enumeration of Some Labelled Trees

Cedric Chauve, Serge Dulucq and Olivier Guibert
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Cedric Chauve: LaBRI, Université Bordeaux I
Serge Dulucq: LaBRI, Université Bordeaux I
Olivier Guibert: LaBRI, Université Bordeaux I

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 146-157 from Springer

Abstract: Abstract In this paper1 we are interested in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let T n,k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of |T n+1,K | = ( k n )n n−k . Moreover, we use the family T n+1,0 to give combinatorial explanations of various identities involving n n . We relate 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 |T n+1,0| = n n to prove bijectively that there are 2n n ordered alternating trees on [n+1].

Keywords: Cayley Tree; Label Tree; Stirling Number; Combinatorial Proof; Combinatorial Interpretation (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-04166-6_13

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DOI: 10.1007/978-3-662-04166-6_13

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