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How many dead ends in a derangement?

Published online by Cambridge University Press:  23 January 2015

Barry Lewis
Barry Lewis*
Affiliation:
110 Highgate Hill, London N6 5HE
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Extract

This article is concerned with two sequences and the matrices they induce. The first, the Derangement sequence is well known to readers of the Gazette. It has been written about and its properties derived in many ways and on many previous occasions [1 – 11]. In contrast the second sequence, the Deadend sequence, is not well known – and certainly not the name since I made that up while writing this article. However, I want to explore these sequences in a single, systematic way. For the former it gives a fresh account of its properties, and for the latter it sets out corresponding results. The surprise is that they tum out to be closely related; surprising since their very different definitions mask such a connection. More than this, most of these properties may be established in the most fundamental way, based on elegant enumerative arguments. So they are free of specialised expertise or technique and may be enjoyed just as they are.

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Articles
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The Mathematical Gazette , Volume 97 , Issue 538 , March 2013, pp. 81 - 94
Copyright
Copyright © The Mathematical Association 2013

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References

1. Goodstein, R. L., How unlikely, Math. Gaz., 23 (May 1939) pp. 205–206.Google Scholar
2. Piggott, H. E., A question of hats, Math. Gaz., 33 (October 1949) pp. 214–215.Google Scholar
3. Rushton, S., On sub-factorial n , Math. Gaz. 34 (December 1950) pp. 302–303.Google Scholar
4. Horsley, S. G., On sub-factorial n , Math. Gaz. 36 (December 1952) p. 285.Google Scholar
5. Biz1ey, M. T. L., A Note on derangements, Math. Gaz., 51 (May 1967) pp. 118–120.Google Scholar
6. Rumney, M., Primrose, E. J. F., A sequence connected with the sub-factorial sequence, Math. Gaz. 52 (December 1968) pp. 381–382.Google Scholar
7. Robin, A. C., The wife-swapping distribution, Math. Gaz., 69 (October 1985) pp. 175–178.Google Scholar
8. Anstice, J., The worst postman problem, Math. Gaz. 72 (October 1988) pp. 226–228.Google Scholar
9. Dennett, J. R., Cricket and derangements, Math. Gaz. 74 (March 1990) pp. 2–5.Google Scholar
10. Lewis, B., Cycles, bicycles, tricycles and more, Math. Gaz., 89 (November 2005) pp. 392–402.CrossRefGoogle Scholar
11. Stephenson, P., Derangements' relation visualised, Math. Gaz. 90 (November 2006) pp. 324–326.Google Scholar
12. Camina, A., Lewis, B., An introduction to enumeration, Springer-Verlag, London (2012) pp. 84–88.Google Scholar