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Mathematical Proceedings of the Cambridge Philosophical Society Laguerre polynomials and derangements
D. Jackson, I. Goulden (1981)
The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian MethodsCanadian Journal of Mathematics, 33
(1964)
Tutte, "The number of plane planted trees with a given partition
A CORRESPONDENCE BETWEEN PLANE PLANTED CHROMATIC TREES AND GENERALISED DERANGEMENTS I. P. GOULDE N AND D. M. JACKSON §1. The correspondence A tree in which adjacent vertices are of different colours is called a chromatic tree. If L is a matrix with element l equal to the number of non-roo t vertices of colour i tj and degree j in a chromatic tree t, then L is called the chromatic partition of t. We demonstrate the existence of a remarkable and tantalising correspondence between two apparently unrelated combinatorial sets, namely plane planted chromatic trees and generalised derangements. The derangement may, of course, be used to construct a graph in which no two vertices of the same colour are joined by an edge, but we cannot guarantee that this graph is in fact a tree. Indeed, we have been unable to give a combinatorial characterisation of this one-to-one correspondence. The correspondence allows us to enumerate K-chromatic trees with given chromatic partition. This is a new result. The case K - 2 has been given previously by Tutte [4]. Throughout this paper we use the following notation: the number of vertices of colour / is n, = £ l ; the
Bulletin of the London Mathematical Society – Wiley
Published: Jan 1, 1981
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