Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T05:53:28.391Z Has data issue: false hasContentIssue false

Enumeration schemes for words avoiding permutations

Published online by Cambridge University Press:  05 October 2010

By
Lara Pudwell
Edited by
Steve Linton ,
Nik Ruškuc  and
Vincent Vatter
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
Get access

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 = w1 … wn. 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
[3] A., Burstein. Enumeration of Words with Forbidden Patterns. PhD thesis, University of Pennsylvania, 1998.
[4] G., Firro. Distanced patterns. PhD thesis, University of Haifa, 2007.
[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
[8] V., Vatter. Enumeration schemes for restricted permutations. Combin. Probab. Comput., 17:137–159, 2008.Google Scholar
[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

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×