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Enumeration schemes for words avoiding permutations

Published online by Cambridge University Press:  05 October 2010

Lara Pudwell
Affiliation:
Department of Mathematics and Computer Science Valparaiso University Valparaiso, IN 46383 USA
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

The enumeration of permutation classes has been accomplished with a variety of techniques. One wide-reaching method is that of enumeration schemes, introduced by Zeilberger and extended by Vatter. In this paper we further extend the method of enumeration schemes to words avoiding permutation patterns. The process of finding enumeration schemes is programmable and allows for the automatic enumeration of many classes of pattern-avoiding words.

Background

The enumeration of permutation classes has been accomplished by many beautiful techniques. One natural extension of permutation classes is pattern-avoiding words. Our concern in this paper is not attractive methods for counting individual classes, but rather developing a systematic technique for enumerating many classes of words. Four main techniques with wide success exist for the systematic enumeration of permutation classes. These are generating trees, insertion encoding, substitution decomposition, and enumeration schemes. In this paper we adapt the method of enumeration schemes, first introduced for permutations by Zeilberger and extended by Vatter to the case of enumerating pattern-restricted words.

Definition 1.1. Let [k]n denote the set of words of length n in the alphabet {1, …, k}, and let w ∈ [k]n, w = w1wn. The reduction of w, denoted by red(w), is the unique word of length n obtained by replacing the ith smallest entries of w with i, for each i.

Type
Chapter
Information
Permutation Patterns , pp. 193 - 212
Publisher: Cambridge University Press
Print publication year: 2010

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References

[1] M. H., Albert, R. E. L., Aldred, M. D., Atkinson, C., Handley, and D., Holton. Permutations of a multiset avoiding permutations of length 3. European J. Combin., 22(8):1021–1031, 2001.Google Scholar
[2] P., Brändén and T., Mansour. Finite automata and pattern avoidance in words. J. Combin. Theory Ser. A, 110(1):127–145, 2005.Google Scholar
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[5] G., Firro and T., Mansour. Three-letter-pattern-avoiding permutations and functional equations. Electron. J. Combin., 13(1):Research Paper 51, 14 pp., 2006.Google Scholar
[6] G., Firro and T., Mansour. Restricted k-ary words and functional equations. Discrete Appl. Math., 157(4):602–616, 2009.Google Scholar
[7] L., Pudwell. Enumeration schemes for words avoiding patterns with repeated letters. Integers, 8:A40, 19, 2008.Google Scholar
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[9] D., Zeilberger. Enumeration schemes and, more importantly, their automatic generation. Ann. Comb., 2(2):185–195, 1998.Google Scholar
[10] D., Zeilberger. On Vince Vatter's brilliant extension of Doron Zeilberger's enumeration schemes for Herb Wilf's classes. The Personal Journal of Ekhad and Zeilberger, published electronically at http://www.math.rutgers.edu/∼zeilberg/pj.html, 2006.Google Scholar
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