Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-18T13:14:29.920Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Bruijn, N. G. de and van Aardenne-Ehrenfest, T., Circuits and trees in oriented linear graphs, Simon Stevin 28 (1959), 203–217.Google Scholar
2. Good, I. J., Generalizations to several variables of Lagrange1 s expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367–380.Google Scholar
3. Good, I. J., The generalization of Lagrange's expansion and the enumeration of trees, Proc. Cambridge Philos. Soc. 61 (1965), 499–517.Google Scholar
4. Goulden, I. P. and Jackson, D. M., A correspondence between plane rooted chromatic trees and generalised derangements, Bull. London Math. Soc. 13 (1981), 28–32.Google Scholar
5. Goulden, I. P. and Jackson, D. M., The enumeration of directed closed Euler trails and directed Hamiltonian circuits by Lagrangian methods, European J. Combinatorics 2 (1981), 131–135.Google Scholar
6. Goulden, I. P. and Jackson, D. M., The generalisation of Tutte's result for chromatic trees, by Lagrangian methods, Can. J. Math. 33 (1981), 12–19.Google Scholar
7. Knuth, D. E., Another enumeration of trees, Can. J. Math. 20 (1968), 1077–1086.Google Scholar
8. Tutte, W. T., The dissection of equilateral triangles into equilateral triangles, Proc. Cambridge Philos. Soc. 44 (1948), 463–482.Google Scholar
9. Tutte, W. T., The number of planted plane trees with a given partition, Amer. Math. Monthly 71 (1964), 272–277.Google Scholar
10. Tutte, W. T., On elementary calculus and the Good formula, J. Combinatorial Theory (B) 18 (1975), 97–137.Google Scholar