Abstract
The 2-adic valuations of Bell and complementary Bell numbers are determined. The complementary Bell numbers are known to be zero at n = 2 and H. S. Wilf conjectured that this is the only case where vanishing occurs. N. C. Alexander and J. An proved (independently) that there are at most two indices where this happens. This paper presents yet an alternative proof of the latter.
To Herb Wilf, with admiration and gratitude
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. C. Alexander. Non-vanishing of Uppuluri-Carpenter. Preprint. 2006.
T. Amdeberhan, D. Manna, and V. Moll. The 2-adic valuation of a sequence arising from a rational integral. Jour. Comb. A, 115:1474–1486, 2008.
T. Amdeberhan, D. Manna, and V. Moll. The 2-adic valuation of Stirling numbers. Experimental Mathematics, 17:69–82, 2008.
T. Amdeberhan, L. Medina, and V. Moll. Arithmetical properties of a sequence arising from an arctangent sum. Journal of Number Theory, 128:1808–1847, 2008.
J. An. Wilf conjecture. Preprint. 2008.
G. P. Egorychev and E. V. Zima. Integral representation and algorithms for closed form summation. In M. Hazewinkel, editor, Handbook of Algebra, volume 5, pages 459–529. Elsevier, 2008.
M. Klazar. Bell numbers, their relatives and algebraic differential equations. J. Comb. Theory Ser. A, 102:63–87, 2003.
M. Klazar. Counting even and odd partitions. Amer. Math. Monthly, 110:527–532, 2003.
A. M. Legendre. Théorie des Nombres. Firmin Didot Frères, Paris, 1830.
V. Moll and X. Sun. A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral. INTEGERS, 10:211–222, 2010.
M. Ram Murty and S. Sumner. On the p-adic series \(\sum _{n=1}^{\infty }{n}^{k} \cdot n!\). In H. Kisilevsky and E. Z. Goren, editors, CRM Proceedings and Lectures Notes. Number Theory, Canadian Number Theory Association VII, pages 219–228. Amer. Math. Soc., 2004.
M. Petkovsek, H. Wilf, and D. Zeilberger. A=B. A. K. Peters, Ltd., 1st edition, 1996.
T. Laffey S. De Wannemacker and R. Osburn. On a conjecture of Wilf. Journal of Combinatorial Theory Series A, 114:1332–1349, 2007.
M. V. Subbarao and A. Verma. Some remarks on a product expansion: An unexplored partition function. In F. G. Garvan and M. E. H. Ismail, editors, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Gainsville, Florida, 1999, pages 267–283. Kluwer, Dordrecht, 2001.
X. Sun and V. Moll. The p-adic valuation of sequences counting alternating sign matrices. Journal of Integer Sequences, 12:09.3.8, 2009.
V. R. Rao Uppuluri and J. A. Carpenter. Numbers generated by the function exp(1 − e x). Fib. Quart., 7:437–448, 1969.
Y. Yang. On a multiplicative partition function. Elec. Jour. Comb., 8:Research Paper 19, 2001.
Acknowledgements
The authors wish to thank the referees for a careful reading of the paper. The last author acknowledges the partial support of NSF-DMS 0713836.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Amdeberhan, T., De Angelis, V., Moll, V.H. (2013). Complementary Bell Numbers: Arithmetical Properties and Wilf’s Conjecture. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-30979-3_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30978-6
Online ISBN: 978-3-642-30979-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)