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-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T23:37:39.725Z Has data issue: false hasContentIssue false

On three different notions of monotone subsequences

Published online by Cambridge University Press:  05 October 2010

By
Miklós Bóna
Edited by
Steve Linton ,
Nik Ruškuc  and
Vincent Vatter
Miklós Bóna
Affiliation:
Department of Mathematics University of Florida Gainesville, FL 32611 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

We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions.

Introduction

Monotone subsequences in a permutation p = p1p2 … pn have been the subject of vigorous research for over sixty years. In this paper, we will review three different lines of work. In all of them, we will consider increasing subsequences of a permutation of length n that have a fixed length k. This is in contrast to another line of work, started by Ulam more than sixty years ago, in which the distribution of the longest increasing subsequence of a random permutation has been studied. That direction of research has recently reached a high point in the article of Baik, Deift, and Johansson.

The three directions we consider are distinguished by their definition of monotone subsequences. We can simply require that k entries of a permutation increase from left to right, or we can in addition require that these k entries be in consecutive positions, or we can even require that they be consecutive integers and be in consecutive positions.

Type
Chapter
Information
Permutation Patterns , pp. 89 - 114
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, M., Elder, A., Rechnitzer, P., Westcott, and M., Zabrocki. On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia. Adv. in Appl. Math., 36(2):95–105, 2006.Google Scholar
[2] R., Arratia. On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern. Electron. J. Combin., 6:Note, N1, 4 pp., 1999.Google Scholar
[3] E., Babson and E., Steingrímsson. Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin., 44:Article B44b, 18 pp., 2000.Google Scholar
[4] J., Baik, P., Deift, and K., Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc., 12(4):1119–1178, 1999.Google Scholar
[5] M., Bóna. Exact and asymptotic enumeration of permutations with subsequence conditions. PhD thesis, M.I.T., 1997.
[6] M., Bóna. Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps. J. Combin. Theory Ser. A, 80(2):257–272, 1997.Google Scholar
[7] M., Bóna. Combinatorics of permutations. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004.
[8] M., Bóna. Generalized descents and normality. Electron. J. Combin., 15(1):Note 21, 8, 2008.Google Scholar
[9] M., Bóna. Where the monotone pattern (mostly) rules. Discrete Math., 308(23):5782–5788, 2008.Google Scholar
[10] S., Elizalde. Asymptotic enumeration of permutations avoiding generalized patterns. Adv. in Appl. Math., 36(2):138–155, 2006.Google Scholar
[11] S., Elizalde and M., Noy. Consecutive patterns in permutations. Adv. in Appl. Math., 30(1–2):110–125, 2003.Google Scholar
[12] J., Fulman. Stein's method and non-reversible Markov chains. In Stein's method: expository lectures and applications, volume 46 of IMS Lecture Notes Monogr. Ser., pages 69–77. Inst. Math. Statist., Beachwood, OH, 2004.Google Scholar
[13] I. M., Gessel. Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A, 53(2):257–285, 1990.Google Scholar
[14] D. M., Jackson and R. C., Read. A note on permutations without runs of given length. Aequationes Math., 17(2–3):336–343, 1978.Google Scholar
[15] D. M., Jackson and J. W., Reilly. Permutations with a prescribed number of p-runs. Ars Combinatoria, 1(1):297–305, 1976.Google Scholar
[16] S., Janson. Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Probab., 16(1):305–312, 1988.Google Scholar
[17] I., Kaplansky. The asymptotic distribution of runs of consecutive elements. Ann. Math. Statistics, 16:200–203, 1945.Google Scholar
[18] A., Marcus and G., Tardos. Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.Google Scholar
[19] A. N., Myers. Counting permutations by their rigid patterns. J. Combin. Theory Ser. A, 99(2):345–357, 2002.Google Scholar
[20] A., Regev. Asymptotic values for degrees associated with strips of Young diagrams. Adv. in Math., 41(2):115–136, 1981.Google Scholar
[21] J., Riordan. Permutations without 3-sequences. Bull. Amer. Math. Soc., 51:745–748, 1945.Google Scholar
[22] A., Ruciński. Proving normality in combinatorics. In Random graphs, Vol. 2 (Poznań, 1989), Wiley-Intersci. Publ., pages 215–231. Wiley, New York, 1992.Google Scholar
[23] R., Simion and F. W., Schmidt. Restricted permutations. European J. Combin., 6(4):383–406, 1985.Google Scholar
[24] R., von Mises. über die wahrscheinlichkeit seltener ereignisse. Z. Angew. Math. Mech., 1(2):121–124, 1921.Google Scholar
[25] R., Warlimont. Permutations avoiding consecutive patterns. Ann. Univ. Sci. Budapest. Sect. Comput., 22:373–393, 2003.Google Scholar
[26] R., Warlimont. Permutations avoiding consecutive patterns. II. Arch. Math. (Basel), 84(6):496–502, 2005.Google Scholar
[27] J., West. Permutations with forbidden subsequences and stack-sortable permutations. PhD thesis, M.I.T., 1990.
[28] J., Wolfowitz. Note on runs of consecutive elements. Ann. Math. Statistics, 15:97–98, 1944.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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
×