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Permuting machines and permutation patterns

Published online by Cambridge University Press:  05 October 2010

Mike Atkinson
Affiliation:
Department of Computer Science University of Otago Dunedin New Zealand
Steve Linton
Affiliation:
University of St Andrews, Scotland
Nik Ruškuc
Affiliation:
University of St Andrews, Scotland
Vincent Vatter
Affiliation:
Dartmouth College, New Hampshire
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Summary

Abstract

Permuting machines were the early inspiration of the theory of permutation pattern classes. Some examples are given which lead up to distilling the key properties that link them to pattern classes. It is shown how relatively simple ways of combining permuting machines can lead to quite complex behaviour and that the notion of regularity can sometimes be used to contain this complexity. Machines which are sensitive to their input data values are shown to be connected to a more general notion than pattern classes. Finally some open problems are presented.

Introduction

Although permutation patterns have only recently been studied in a systematic manner their history can be traced back many decades. It could be argued that the well-known lemma of Erdős and Szekeres is really a result about pattern classes (a pattern class whose basis contains both an increasing and a decreasing permutation is necessarily finite). However it is perhaps more convincing to attribute the birth of the subject to the ground-breaking first volume of Donald Knuth's Art of Computer Programming series. In the main body of his text, and in some fascinating follow-up exercises, Knuth enumerated some pattern classes, and found some bases, while at the same time introducing some techniques on generating functions that, in due time, were codified as “the kernel method”.

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Chapter
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Permutation Patterns , pp. 67 - 88
Publisher: Cambridge University Press
Print publication year: 2010

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References

[1] M. H., Albert, R. E. L., Aldred, M. D., Atkinson, H. P., van Ditmarsch, C. C., Handley, D. A., Holton, and D. J., McCaughan. Compositions of pattern restricted sets of permutations. Australas. J. Combin., 37:43–56, 2007.Google Scholar
[2] M. H., Albert, M. D., Atkinson, and S., Linton. Permutations generated by stacks and deques. Ann. Comb., 14(1):3–16, 2010.Google Scholar
[3] M. H., Albert, S., Linton, and N., Ruškuc. On the permutational power of token passing networks. In this volume, 317–338.
[4] M. H., Albert, S., Linton, and N., Ruškuc. The insertion encoding of permutations. Electron. J. Combin., 12(1):Research paper 47, 31 pp., 2005.Google Scholar
[5] R. E. L., Aldred, M. D., Atkinson, H. P., van Ditmarsch, C. C., Handley, D. A., Holton, and D. J., McCaughan. Permuting machines and priority queues. Theoret. Comput. Sci., 349(3):309–317, 2005.Google Scholar
[6] M. D., Atkinson, M. J., Livesey, and D., Tulley. Permutations generated by token passing in graphs. Theoret. Comput. Sci., 178(1–2):103–118, 1997.Google Scholar
[7] M. D., Atkinson and M., Thiyagarajah. The permutational power of a priority queue. BIT, 33(1):2–6, 1993.Google Scholar
[8] A., Björner. Orderings of Coxeter groups. In Combinatorics and algebra (Boulder, Colo., 1983), volume 34 of Contemp. Math., pages 175–195. Amer. Math. Soc., Providence, RI, 1984.Google Scholar
[9] M., Bóna. A survey of stack-sorting disciplines. Electron. J. Combin., 9(2):Article 1, 16 pp., 2003.Google Scholar
[10] T., Chow and J., West. Forbidden subsequences and Chebyshev polynomials. Discrete Math., 204(1–3):119–128, 1999.Google Scholar
[11] M., Elder. Permutations generated by a stack of depth 2 and an infinite stack in series. Electron. J. Combin., 13:Research paper 68, 12 pp., 2006.Google Scholar
[12] P., Erdős and G., Szekeres. A combinatorial problem in geometry. Compos. Math., 2:463–470, 1935.Google Scholar
[13] S., Even and A., Itai. Queues, stacks, and graphs. In Theory of machines and computations (Proc. Internat. Sympos., Technion, Haifa, 1971), pages 71–86. Academic Press, New York, 1971.Google Scholar
[14] J. D., Gilbey and L. H., Kalikow. Parking functions, valet functions and priority queues. Discrete Math., 197/198:351–373, 1999. 16th British Combinatorial Conference (London, 1997).Google Scholar
[15] D. E., Knuth. The art of computer programming. Vol. 1: Fundamental algorithms. Addison-Wesley Publishing Co., Reading, Mass., 1969.Google Scholar
[16] M. M., Murphy. Restricted Permutations, Antichains, Atomic Classes, and Stack Sorting. PhD thesis, Univ. of St Andrews, 2002.
[17] V. R., Pratt. Computing permutations with double-ended queues, parallel stacks and parallel queues. In STOC '73: Proceedings of the fifth annual ACM symposium on Theory of computing, pages 268–277, New York, NY, USA, 1973. ACM Press.Google Scholar
[18] P., Rosenstiehl and R. E., Tarjan. Gauss codes, planar Hamiltonian graphs, and stack-sortable permutations. J. Algorithms, 5(3):375–390, 1984.Google Scholar
[19] R., Simion and F. W., Schmidt. Restricted permutations. European J. Combin., 6(4):383–406, 1985.Google Scholar
[20] R., Tarjan. Sorting using networks of queues and stacks. J. Assoc. Comput. Mach., 19:341–346, 1972.Google Scholar
[21] S., Waton. On Permutation Classes Defined by Token Passing Networks, Gridding Matrices and Pictures: Three Flavours of Involvement. PhD thesis, Univ. of St Andrews, 2007.

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