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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

F., Bergeron, G., Labelle and P., Leroux, Combinatorial Species and Tree-like Structures, Encyclopedia of Mathematics and its Applications 67, Cambridge University Press, Cambridge, 1998.
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.
P., Flajolet and R., Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.
R., Fraïssé, Theory of Relations, North-Holland, Amsterdam, 1986.
M. R., Garey and D. S., Johnson, Computers and Intractability: An Introduction to the Theory of NP-completeness, W. H. Freeman, San Francisco, 1979.
I. P., Goulden and D. M., Jackson, Combinatorial Enumeration,Wiley-Interscience, New York, 1983.
Wilfrid|Hodges, A Shorter Model Theory, Cambridge University Press, Cambridge, 1997.
V., Kac and P., Cheung, Quantum Calculus, Springer, New York, 2002.
Sergey, Kitaev, Patterns in permutations and words, Monographs in Theoretical Computer Science, Springer-Verlag, 2011.
J. H. van, Lint and R. M., Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 2001.
I. G., Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 1999.
Marko, Petkovsek, Herbert S., Wilf and Doron, Zeilberger, A = B, A. K. Peters Ltd., Wellesley, MA, 1996.
A., Slomson, An Introduction to Combinatorics, Chapman and Hall, London, 1991.
R. P., Stanley, Enumerative Combinatorics (2 volumes), Cambridge University Press, Cambridge, 2000, 2001.
E. A., Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), 485– 515.Google Scholar
A. T., Benjamin, G. M., Levin, K., Mahlburg and J. J., Quinn, Random approaches to Fibonacci identities, Amer. Math. Monthly 107 (2000), 511–516.Google Scholar
N., Boston, W., Dabrowski, T., Foguel, P. J., Gies, J., Leavitt, D. T., Ose and D. A., Jackson, The proportion of fixed-point-free elements of a transitive permutation group, Commun. Algebra 21 (1993), 3259–3275.Google Scholar
P. J., Cameron, Oligomorphic permutation groups, in Perspectives in Mathematical Sciences II (ed. N. S., Narasimha Sastry, T. S. S. R. K., Rao, Mohan, Delampady and B., Rajeev), World Scientific, Singapore, 2009, pp. 37–61.
P. J., Cameron and A. M., Cohen, On the number of fixed point free elements of a permutation group, Discrete Math. 106/107 (1992), 135–138.Google Scholar
P. J., Cameron, D., Gewurz and F., Merola, Product action, Discrete Math. 308 (2008), 386–394.Google Scholar
P. J., Cameron, B., Jackson and J. D., Rudd, An orbital Tutte polynomial for graphs and matroids, Discrete Math. 308 (2008), 920–930.Google Scholar
P. J., Cameron, T., Prellberg and D., Stark, Asymptotic enumeration of incidence matrices, Journal of Physics: Conference Series 42 (2006), 59–70.Google Scholar
A., Dress and T., Müller, Decomposable functors and the exponential principle, Advances in Mathematics 129 (1997), 188–221.Google Scholar
S., Eberhard, K., Ford and B., Green, Permutations fixing a k-set, https://arxiv.org/abs/1507.04465
W. T., Gowers, The two cultures of mathematics, pp. 65–78 in Mathematics: Frontiers and Perspectives (ed. V., Arnold, M., Atiyah, P., Lax and B., Mazur), American Math. Soc., Providence, 1999.
L. J., Guibas and A. M., Odlyzko, String overlaps, pattern matching, and nontransitive games, J. Combinatorial Theory (A) 30 (1981), 183–208.Google Scholar
W. K., Hayman, A generalization of Stirling's formula, J. Reine Angew. Math. 196 (1956), 67–95.Google Scholar
A., Joyal, Une theorie combinatoire des séries formelles, Advances in Math. 42 (1981), 1–82.Google Scholar
M., Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux 9 (1997), 221–228.Google Scholar
A., Laradji and A., Umar, Combinatorial results for the symmetric inverse semigroup, Semigroup Forum 75 (2007), 221–236.Google Scholar
R., Lewis and S. P., Norton, On a problem raised by P. J. Cameron, Discrete Mathematics 138 (1995), 315–318.Google Scholar
N., Linial and N., Nisan, Approximate inclusion-exclusion, Combinatorica 10 (1990), 349–365.Google Scholar
S., Majid, Free braided differential calculus, braided binomial theorem and the braided exponential map, J. Math. Phys. 34 (1993), 4843–4856.Google Scholar
F., Merola, Orbits on n-tuples for infinite permutation groups, Europ. J. Combinatorics 22 (2001), 225–241.Google Scholar
Peter M., Neumann, A lemma that is not Burnside's, The Mathematical Scientist 4 (1979), 133–141.Google Scholar
Maurice, Pouzet, When is the orbit algebra of a group an integral domain? Proof of a conjecture of P. J. Cameron, Theor. Inform. Appl. 42 (2008), 83–103.Google Scholar
A. D., Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in Surveys in Combinatorics 2005 (ed. B. S., Webb), London Math. Soc. Lecture Notes, 327, Cambridge Univ. Press, Cambridge, 2005, pp. 173–226.
R. P., Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178.Google Scholar
H. S., Wilf, The ‘Snake Oil’ method for proving combinatorial identities, in Surveys in Combinatorics (ed. J., Siemons), London Math. Soc. Lecture Note Series 141, Cambridge University Press, Cambridge, 1989, pp. 208–217.
Doron, Zeilberger, Automatic counting of tilings of skinny plane regions, in Surveys in Combinatorics 2013 (ed. Simon R., Blackburn, Stefanie, Gerke and Mark, Wildon), London Math. Soc. Lecture Note Series 409, Cambridge University Press, Cambridge, 2013, pp. 363–378.

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