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
×
Home
Hostname: page-component-79b67bcb76-vkbph Total loading time: 0.341 Render date: 2021-05-14T04:16:39.229Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }
EnglishFrançais
Mathematika Mathematika

Article contents

ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS

Part of: Combinatorics

Published online by Cambridge University Press:  14 August 2019

Robert Brignall ,
Vít Jelínek ,
Jan Kynčl  and
David Marchant
Robert Brignall
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, U.K. email robert.brignall@open.ac.uk
Vít Jelínek
Affiliation:
Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic email jelinek@iuuk.mff.cuni.cz
Jan Kynčl
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic email kyncl@kam.mff.cuni.cz
David Marchant
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, U.K. email david.marchant@open.ac.uk
Corresponding
E-mail address:
robert.brignall@open.ac.uk
jelinek@iuuk.mff.cuni.cz
kyncl@kam.mff.cuni.cz
david.marchant@open.ac.uk
Get access
Rights & Permissions[Opens in a new window]

Abstract

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$ . This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations  $\unicode[STIX]{x1D70B}$ . We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$ , including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$ .

MSC classification

Primary: 05A05: Permutations, words, matrices
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 1074 - 1092
Copyright
Copyright © University College London 2019 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

Footnotes

V. Jelínek and J. Kynčl were supported by project 16-01602Y of the Czech Science Foundation (GAČR). J. Kynčl was also supported by Charles University project UNCE/SCI/004.

References

Albert, M. H. and Atkinson, M. D., Simple permutations and pattern restricted permutations. Discrete Math. 300(1-3) 2005, 1–15.10.1016/j.disc.2005.06.016CrossRefGoogle Scholar
Albert, M. H., Atkinson, M. D. and Klazar, M., The enumeration of simple permutations. J. Integer Seq. 6(4) 2003, Article 03.4.4.Google Scholar
Atkinson, M. D. and Stitt, T., Restricted permutations and the wreath product. Discrete Math. 259(1–3) 2002, 19–36.10.1016/S0012-365X(02)00443-0CrossRefGoogle Scholar
Björner, A., The Möbius function of subword order. In Invariant Theory and Tableaux (Minneapolis, MN, 1988) (IMA Vol. Math. Appl., 19 ), Springer (New York, 1990), 118–124.Google Scholar
Brignall, R., Jelínek, V., Kynčl, J. and Marchant, D., Zeros of the Möbius function of permutations. Preprint, 2018, arXiv:org/abs/1810.05449v1.Google Scholar
Brignall, R. and Marchant, D., The Möbius function of permutations with an indecomposable lower bound. Discrete Math. 341(5) 2018, 1380–1391.10.1016/j.disc.2018.02.012CrossRefGoogle Scholar
Burstein, A., Jelínek, V., Jelínková, E. and Steingrímsson, E., The Möbius function of separable and decomposable permutations. J. Combin. Theory Ser. A 118(8) 2011, 2346–2364.10.1016/j.jcta.2011.06.002CrossRefGoogle Scholar
Euler, L., Recherches sur une nouvelle espèce des quarrés magiques. Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen 9 1782, 85–239.Google Scholar
Jelínek, V., Kantor, I., Kynčl, J. and Tancer, M., On the growth of the Möbius function of permutations. J. Combin. Theory, Ser. A 169 2020,105121.Google Scholar
Kaplansky, I., The asymptotic distribution of runs of consecutive elements. Ann. Math. Stat. 16(2) 1945, 200–203.10.1214/aoms/1177731121CrossRefGoogle Scholar
Rumney, M. and Primrose, E. J. F., A sequence connected with the sub-factorial sequence. Gaz. Math. 52(382) 1968, 381–382.10.2307/3611860CrossRefGoogle Scholar
Sagan, B. E. and Vatter, V., The Möbius function of a composition poset. J. Algebraic Combin. 24(2) 2006, 117–136.10.1007/s10801-006-0017-4CrossRefGoogle Scholar
Sloane, N. J. A., The on-line encyclopedia of integer sequences, published electronically at http://oeis.org/.Google Scholar
Smith, J. P., On the Möbius function of permutations with one descent. Electron. J. Combin. 21(2) 2014, 19 Paper 2.11.Google Scholar
Smith, J. P., Intervals of permutations with a fixed number of descents are shellable. Discrete Math. 339(1) 2016, 118–126.10.1016/j.disc.2015.08.004CrossRefGoogle Scholar
Smith, J. P., A formula for the Möbius function of the permutation poset based on a topological decomposition. Adv. Appl. Math. 91 2017, 98–114.10.1016/j.aam.2017.06.002CrossRefGoogle Scholar
Smith, J. P., private correspondence, 2018.Google Scholar
Stanley, R. P., Enumerative Combinatorics, 2nd edn., Vol. 1 (Cambridge Studies in Advanced Mathematics 49 ), Cambridge University Press (Cambridge, 2012).Google Scholar
Steingrímsson, E. and Tenner, B. E., The Möbius function of the permutation pattern poset. J. Comb. 1(1) 2010, 39–52.Google Scholar
Wilf, H. S., The patterns of permutations. Discrete Math. 257(2-3) 2002, 575–583.10.1016/S0012-365X(02)00515-0CrossRefGoogle Scholar
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and 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 <service> account. Find out more about sending content to Dropbox.

ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and 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 <service> account. Find out more about sending content to Google Drive.

ZEROS OF THE MÖBIUS FUNCTION OF PERMUTATIONS
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *