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The Application of Lagrangian Methods to the Enumeration of Labelled Trees with Respect to Edge Partition

Published online by Cambridge University Press:  20 November 2018

I. P. Goulden  and
D. M. Jackson
I. P. Goulden
Affiliation:
University of Waterloo, Waterloo, Ontario
D. M. Jackson
Affiliation:
University of Waterloo, Waterloo, Ontario
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In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.

In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 513 - 518
Copyright
Copyright © Canadian Mathematical Society 1982

References

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