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Sequence Enumeration and the de Bruijn-Van Aardenne Ehrenfest-Smith-Tutte Theorem

Published online by Cambridge University Press:  20 November 2018

D. M. Jackson  and
I. P. Goulden
D. M. Jackson
Affiliation:
University of Waterloo, Waterloo, Ontario
I. P. Goulden
Affiliation:
University of Waterloo, Waterloo, Ontario
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The de Bruijn—van Aardenne Ehrenfest— Smith—Tutte theorem [1] is a theorem which connects the number of Eulerian dicircuits in a directed graph with the number of rooted spanning arborescences. In this paper we obtain a proof of this theorem by considering sequences over a finite alphabet, and we show that the theorem emerges from the generating function for a certain type of sequence. The generating function for the set of sequences is obtained as the solution of a linear system of equations in Section 2. The power series expansion for the solution of this system is obtained by means of the multivariate form of the Lagrange theorem for implicit functions, and is given in Section 3, together with a restatement of the theorem as a matrix identity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 488 - 495
Copyright
Copyright © Canadian Mathematical Society 1979

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