Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-15T07:47:45.552Z Has data issue: false hasContentIssue false

A Formal Solution of Σi=1AieBix = X

Published online by Cambridge University Press:  20 November 2018

George N. Raney*
Affiliation:
University of Connecticut Storrs, Conn.
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.

We consider a combinatorial enumeration problem involving certain collections of labelled rooted trees having coloured nodes and edges. Notations and definitions are introduced in §§2 and 3, and the problem is described in §4. We give recursion formulas for its solution in §5. Then, by using a modification of a method of Priifer, we obtain a direct solution in terms of multinomial coefficients and power products in §§6 and 7. These results are combined in §8. Working in a formal power series algebra we find a formal solution of the equation

which expresses the unknown X as a multiple power series in the Ai and the Bi.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Clarke, L. E., On Cayley's formula for counting trees, J. Lond. Math. Soc, 33 (1958), 471474»Google Scholar
2. Neville, E. H., The codifying of tree-structure, Proc. Camb. Phil. Soc, 49 (1953), 381385.Google Scholar
3. Prüfer, H., Neuer Beweis eines Satzes über Permutationen, Arch. Math. u. Phys., 27 (1918). 142144.Google Scholar