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13 - Bibliography and further directions

Published online by Cambridge University Press:  28 June 2017

Peter J. Cameron
Peter J. Cameron
Affiliation:
University of St Andrews, Scotland
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Summary

In this chapter I list a few books, papers and websites which may be useful if you would like to follow up some of the things I have discussed.

The On-line Encyclopedia of Integer Sequences

The On-line Encyclopedia of Integer Sequences, available at the URL https://oeis.org/,

is an essential resource for anyone doing research in combinatorics. For example, suppose you are trying to count the number of arrangements of n zeros and ones around a circle in which no two ones are consecutive (Exercise 2.8(b)). You might reasonably assume that n ≥ 3, and calculate that for n = 3,4,5 there are respectively 4, 7 and 11 such arrangements. If you type these three numbers into the Encyclopedia, you find many matches (I found 521 when I tried it on 24 October 2016), but near the top (indeed, at the top when I did the experiment) is an entry for the Lucas numbers. The entry gives a recurrence relation (identical to that for Fibonacci numbers), congruences modulo primes, representation in terms of hyperbolic functions, and much more, including (most importantly) ten references to the literature, short programs for computing the numbers in various programming languages, further web links, open problems, and cross-references to related sequences. Now you can either prove directly that the numbers you are interested in satisfy the Fibonacci recurrence (and hence coincide with the Lucas numbers), or check in the literature for further information which will help you make the identification.

On the Encyclopedia's website, you will find different ways of viewing the sequence and information about it, pointers to interesting or mysterious sequences,

a formula for the terms of the sequence or its generating function if known, and several articles by the editor Neil Sloane and others describing uses of the Encyclopedia in research. I have used it myself on a number of occasions.

Type
Chapter
Information
Notes on Counting: An Introduction to Enumerative Combinatorics , pp. 212 - 216
Publisher: Cambridge University Press
Print publication year: 2017

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References

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P. J., Cameron, Oligomorphic Permutation Groups, London Math. Soc. Lecture Notes 152, Cambridge University Press, Cambridge, 1990.
P. J., Cameron, Permutation Groups, London Math. Soc. Student Texts 45, Cambridge University Press, Cambridge, 1999.
J. H., Conway and R. K., Guy, The Book of Numbers, Springer-Verlag, New York, 1996.
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