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Analysis of Statistics for Generalized Stirling Permutations

Published online by Cambridge University Press:  11 October 2011

MARKUS KUBA  and
ALOIS PANHOLZER
MARKUS KUBA
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria (e-mail: kuba@dmg.tuwien.ac.at, Alois.Panholzer@tuwien.ac.at) HTL Wien 5 Spengergasse, Spengergasse 20, 1050 Wien, Austria
ALOIS PANHOLZER
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria (e-mail: kuba@dmg.tuwien.ac.at, Alois.Panholzer@tuwien.ac.at)
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Abstract

In this work we give a study of generalizations of Stirling permutations, a restricted class of permutations of multisets introduced by Gessel and Stanley [15]. First we give several bijections between such generalized Stirling permutations and various families of increasing trees extending the known correspondences of [20, 21]. Then we consider several permutation statistics of interest for generalized Stirling permutations as the number of left-to-right minima, the number of left-to-right maxima, the number of blocks of specified sizes, the distance between occurrences of elements, and the number of inversions. For all these quantities we give a distributional study, where the established connections to increasing trees turn out to be very useful. To obtain the exact and limiting distribution results we use several techniques ranging from generating functions, connections to urn models, martingales and Stein's method.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 875 - 910
Copyright
Copyright © Cambridge University Press 2011

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