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ENUMERATION OF A DUAL SET OF STIRLING PERMUTATIONS BY THEIR ALTERNATING RUNS

Part of: Combinatorics

Published online by Cambridge University Press:  01 April 2016

SHI-MEI MA  and
HAI-NA WANG
SHI-MEI MA*
Affiliation:
Department of Mathematics, Northeastern University, Shenyang 110004, China email shimeimapapers@163.com
HAI-NA WANG
Affiliation:
Department of Mathematics, Northeastern University, Shenyang 110004, China email hainawangpapers@163.com
*
✉ shimeimapapers@163.com
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Abstract

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In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

Keywords

alternating runscontext-free grammarsEulerian polynomialsStirling permutations

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions
Secondary: 05A19: Combinatorial identities, bijective combinatorics

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 177 - 186
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Adin, R., Brenti, F. and Roichman, Y., ‘Descent numbers and major indices for the hyperoctahedral group’, Adv. Appl. Math. 27 (2001), 210–224.Google Scholar
André, D., ‘Étude sur les maxima, minima et séquences des permutations’, Ann. Sci. Éc. Norm. Supér. 3(1) (1884), 121–135.Google Scholar
Bóna, M., ‘Real zeros and normal distribution for statistics on Stirling permutations defined by Gessel and Stanley’, SIAM J. Discrete Math. 23 (2008–2009), 401–406.Google Scholar
Canfield, E. R. and Wilf, H., ‘Counting permutations by their alternating runs’, J. Combin. Theory Ser. A 115 (2008), 213–225.Google Scholar
Carlitz, L., ‘The coefficients in an asymptotic expansion’, Proc. Amer. Math. Soc. 16 (1965), 248–252.CrossRefGoogle Scholar
Chen, W. Y. C., ‘Context-free grammars, differential operators and formal power series’, Theoret. Comput. Sci. 117 (1993), 113–129.Google Scholar
Gessel, I. and Stanley, R. P., ‘Stirling polynomials’, J. Combin. Theory Ser. A 24 (1978), 25–33.Google Scholar
Janson, S., Kuba, M. and Panholzer, A., ‘Generalized Stirling permutations, families of increasing trees and urn models’, J. Combin. Theory Ser. A 118 (2011), 94–114.Google Scholar
Kuba, M. and Panholzer, A., ‘Enumeration formulae for pattern restricted Stirling permutations’, Discrete Math. 312 (2012), 3179–3194.Google Scholar
Ma, S.-M., ‘A family of two-variable derivative polynomials for tangent and secant’, Electron. J. Combin. 20(1) (2013), #P11.CrossRefGoogle Scholar
Ma, S.-M., ‘Enumeration of permutations by number of alternating runs’, Discrete Math. 313 (2013), 1816–1822.Google Scholar
Ma, S.-M., ‘Some combinatorial arrays generated by context-free grammars’, European J. Combin. 34 (2013), 1081–1091.CrossRefGoogle Scholar
Ma, S.-M. and Mansour, T., ‘The 1/k-Eulerian polynomials and k-Stirling permutations’, Discrete Math. 338 (2015), 1468–1472.Google Scholar
Ma, S.-M. and Yeh, Y.-N., ‘Stirling permutations, cycle structure of permutations and perfect matchings’, Electron. J. Combin. 22(4) (2015), #P4.42.Google Scholar
Mansour, T. and Schork, M., Commutation Relations, Normal Ordering and Stirling Numbers, Discrete Mathematics and its Applications Series (Chapman and Hall, CRC Press, Taylor and Francis, Boca Raton, FL, 2015).Google Scholar
Riordan, J., ‘The blossoming of Schrø̈der’s fourth problem’, Acta Math. 137(1–2) (1976), 1–16.Google Scholar
Savage, C. D. and Schuster, M. J., ‘Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences’, J. Combin. Theory Ser. A 119 (2012), 850–870.Google Scholar
Savage, C. D. and Viswanathan, G., ‘The 1/k-Eulerian polynomials’, Electron. J. Combin. 19 (2012), #P9.Google Scholar
Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences (2010), http://oeis.org.Google Scholar