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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.

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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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References

[1]Barbour, A. D. and Eagleson, G. K. (1985) Multiple comparisons and sums of dissociated random variables. Adv. Appl. Probab. 17 147–162.CrossRefGoogle Scholar
[2]Barbour, A. D., Karoński, M. and Ruciński, A. (1989) A central limit theorem for decomposable random variables with applications to random graphs, J. Combin. Theory Ser. B 47 125–145.CrossRefGoogle Scholar
[3]Bender, E. A. (1973) Central and local limit theorems applied to asymptotics enumeration. J. Combin. Theory Ser. A 15 91–111.CrossRefGoogle Scholar
[4]Bergeron, F., Flajolet, P. and Salvy, B. (1992) Varieties of increasing trees. In CAAP'1992: Proc. 17th Colloquium on Trees in Algebra and Programming, Vol. 581 of Lecture Notes in Computer Science, Springer, pp. 24–48.Google Scholar
[5]Bollobás, B. and Riordan, O. M. (2003) Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks, Wiley-VCH, pp. 1–34.Google Scholar
[6]Bona, M. (2004) Combinatorics of Permutations, Chapman & Hall.CrossRefGoogle Scholar
[7]Bona, M. (2008) Real zeros and normal distribution for statistics on Stirling permutations defined by Gessel and Stanley. SIAM J. Discrete Math. 23 401–406.CrossRefGoogle Scholar
[8]Brenti, F. (1989) Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81 #413.Google Scholar
[9]Brenti, F. (1998) Hilbert polynomials in combinatorics. J. Algebraic Combin. 7 127–156.CrossRefGoogle Scholar
[10]Broutin, N., Devroye, L., McLeish, E. and de la Salle, M. (2008) The height of increasing trees. Random Struct. Alg. 32 494–518.CrossRefGoogle Scholar
[11]Chan, D. Y. C., Hughes, B. D., Leong, A. S. and Reed, W. J. (2003) Stochastically evolving networks. Phys. Rev. E 68 066124.CrossRefGoogle ScholarPubMed
[12]Eggenberger, F. and Pólya, G. (1923) Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech. 3 279–289.CrossRefGoogle Scholar
[13]Flajolet, P., Dumas, P. and Puyhaubert, V. (2006) Some exactly solvable models of urn process theory. In Discrete Mathematics and Computer Science: Proc. Fourth Colloquium on Mathematics and Computer Science (Chassaing, P., ed.), AG, pp. 59–118.Google Scholar
[14]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[15]Gessel, I. and Stanley, R. P. (1978) Stirling polynomials, J. Combin Theory Ser. A 24 24–33.CrossRefGoogle Scholar
[16]Hwang, H.-K. (1998) On convergence rates in the central limit theorems for combinatorial structures. European J. Combin. 19 329–343.CrossRefGoogle Scholar
[17]Janson, S. (2004) Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.CrossRefGoogle Scholar
[18]Janson, S. (2005) Asymptotic degree distribution in random recursive trees. Random Struct. Alg. 26 69–83.CrossRefGoogle Scholar
[19]Janson, S. (2006) Limit theorems for triangular urn schemes. Probab. Theory Rel. Fields 134 417–452.CrossRefGoogle Scholar
[20]Janson, S. (2008) Plane recursive trees, Stirling permutations and an urn model. DMTCS: Proc. Fifth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp. 541–548.CrossRefGoogle Scholar
[21]Janson, S., Kuba, M. and Panholzer, A. (2011) Generalized Stirling permutations, families of increasing trees and urn models. J. Combin. Theory Ser. A 118 94–114.CrossRefGoogle Scholar
[22]Johnson, N. L. and Kotz, S. (1977) Urn Models and their Application, Wiley.Google Scholar
[23]Kuba, M. and Panholzer, A. (2006) Descendants in increasing trees. Electron. J. Combin. 13 #8.CrossRefGoogle Scholar
[24]Kuba, M. and Panholzer, A. (2007) Limiting distributions for a class of diminishing urn models. Manuscript, (submitted).CrossRefGoogle Scholar
[25]Loève, M. (1977) Probability Theory I, fourth edition, Springer.Google Scholar
[26]Louchard, G. and Prodinger, H. (2003) Inversions in permutations: A saddle point approach. J. Integers Sequences 6 #3.2.8.Google Scholar
[27]Mahmoud, H. (2003) Urn models and connections to random trees: A review. J. Iranian Math. Soc. 2 53–114.Google Scholar
[28]Mahmoud, H. and Smythe, R. (1991) On the distribution of leaves in rooted subtrees of recursive trees. Annals Appl. Probab. 1 406–418.CrossRefGoogle Scholar
[29]Mahmoud, H. M., Smythe, R. T. and Szymański, J. (1993) On the structure of random plane-oriented recursive trees and their branches. Random Struct. Alg. 4 151–176.CrossRefGoogle Scholar
[30]Margolius, B. H. (2001) Permutations with inversions. J. Integer Sequences 4 1–13.Google Scholar
[31]Panholzer, A. and Prodinger, H. (2007) The level of nodes in increasing trees revisited. Random Struct. Alg. 31 203–226.CrossRefGoogle Scholar
[32]Park, S. K. (1994) The r-multipermutations. J. Combin. Theory Ser. A 67 44–71.CrossRefGoogle Scholar
[33]Park, S. K. (1994) Inverse descents of r-multipermutations. Discrete Math. 132 215–229.CrossRefGoogle Scholar
[34]Park, S. K. (1994) P-partitions and q-Stirling numbers. J. Combin. Theory Ser. A 68 33–52.CrossRefGoogle Scholar
[35]Pólya, G. (1931) Sur quelques points de la théorie des probabilités. Ann. Inst. Poincaré 1 117–161.Google Scholar
[36]Prodinger, H. (1996) Descendants in heap ordered trees, or, A triumph of computer algebra. Electron. J. Combin. 3 #29.CrossRefGoogle Scholar
[37]Sachkov, V. N. (1997) Probabilistic Methods in Combinatorial Analysis, Cambridge University Press.CrossRefGoogle Scholar
[38]Vitter, J. and Flajolet, P. (1990) Average case analysis of algorithms and data structures. In Handbook of Theoretical Computer Science, Elsevier, pp. 431–524.Google Scholar
[39]Wilf, H. (1994) Generatingfunctionology, second edition, Academic Press.Google Scholar