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Enumeration of partitions by rises, levels and descents

Published online by Cambridge University Press:  05 October 2010

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
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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.

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Information
Permutation Patterns , pp. 221 - 232
Publisher: Cambridge University Press
Print publication year: 2010

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References

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