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-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-10T22:39:43.858Z Has data issue: false hasContentIssue false

Enumeration of partitions by rises, levels and descents

Published online by Cambridge University Press:  05 October 2010

By
Toufik Mansour  and
Augustine O. Munagi
Edited by
Steve Linton ,
Nik Ruškuc  and
Vincent Vatter
Toufik Mansour
Affiliation:
Department of Mathematics Haifa University 31905 Haifa, Israel
Augustine O. Munagi
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory School of Mathematics University of the Witwatersrand Johannesburg 2050, South Africa
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

A descent in a permutation α1α2 · αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic which was first studied by P. A. MacMahon almost a hundred years ago, and it still plays an important role in the study of permutations. Representing set partitions by equivalent canonical sequences of integers, we study this statistic among the set partitions, as well as the numbers of rises and levels. We enumerate set partitions with respect to these statistics by means of generating functions, and present some combinatorial proofs. Applications are obtained to new combinatorial results and previously-known ones.

Introduction

A descent in a permutation α = α1α2 ··· αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic. This statistic was first studied by MacMahon, and it still plays an important role in the study of permutation statistics. In this paper we study the statistics of numbers of rises, levels and descents among set partitions expressed as canonical sequences, defined below.

Type
Chapter
Information
Permutation Patterns , pp. 221 - 232
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] H., Davenport and A., Schinzel. A combinatorial problem connected with differential equations. Amer. J. Math., 87:684–694, 1965.Google Scholar
[2] R. L., Graham, D. E., Knuth, and O., Patashnik. Concrete mathematics. Addison-Wesley Publishing Company, Reading, MA, second edition, 1994.Google Scholar
[3] V., Jelínek and T., Mansour. On pattern-avoiding partitions. Electron. J. Combin., 15(1):Research paper 39, 52, 2008.Google Scholar
[4] M., Klazar. On abab-free and abba-free set partitions. European J. Combin., 17(1):53–68, 1996.Google Scholar
[5] M., Klazar. On trees and noncrossing partitions. Discrete Appl. Math., 82(1-3):263–269, 1998.Google Scholar
[6] P. A., MacMahon. Combinatory Analysis. Cambridge University Press, London, 1915/16.Google Scholar
[7] R. C., Mullin and R. G., Stanton. A map-theoretic approach to Davenport-Schinzel sequences. Pacific J. Math., 40:167–172, 1972.Google Scholar
[8] B. E., Sagan. Pattern avoidance in set partitions. arXiv:math/0604292v1 [math.CO].
[9] R. P., Stanley. Enumerative combinatorics. Vol. 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.Google Scholar
[10] M., Wachs and D., White. p, q-Stirling numbers and set partition statistics. J. Combin. Theory Ser. A, 56(1):27–46, 1991.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
×