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A Formal Solution of Σ∞i=1AieBix = X

Published online by Cambridge University Press:  20 November 2018

George N. Raney
George N. Raney*
Affiliation:
University of Connecticut Storrs, Conn.
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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
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 755 - 762
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Clarke, L. E., On Cayley's formula for counting trees, J. Lond. Math. Soc, 33 (1958), 471–474»Google Scholar
2. Neville, E. H., The codifying of tree-structure, Proc. Camb. Phil. Soc, 49 (1953), 381–385.Google Scholar
3. Prüfer, H., Neuer Beweis eines Satzes über Permutationen, Arch. Math. u. Phys., 27 (1918). 142–144.Google Scholar