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Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations

Published online by Cambridge University Press:  24 October 2008

P. Flajolet  and
A. M. Odlyzko
P. Flajolet
Affiliation:
INRIA, 78150 Rocquencourt, France
A. M. Odlyzko
Affiliation:
Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A.
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Abstract

This paper studies coefficients yh, n of sequences of polynomials

defined by non-linear recurrences. A typical example to which the results of this paperapply is that of the sequence

which arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficients yh, n as functions of n with h fixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially as h → ∞. Applications of these results include enumerations of binary trees and 2–3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 237 - 253
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

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