A generalized recurrence formula for Stirling numbers and related sequences

Mark Shattuck
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 74—80
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Authors and affiliations

Mark Shattuck
Department of Mathematics, University of Tennessee
37996 Knoxville, TN, USA

Abstract

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.

Keywords

• Stirling numbers
• r-Whitney numbers
• q-Stirling numbers

AMS Classification

• 05A19
• 05A18

References

1. Benjamin, A. T. & J. J. Quinn (2003) Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America.
2. Broder, A. Z. (1984) The r-Stirling numbers, Discrete Math., 49, 241–259.
3. Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., 15, 987–1000.
4. Cheon, G.-S. & J.-H. Jung (2012) r-Whitney numbers of Dowling lattices, Discrete Math., 312, 2337–2348.
5. Graham, R. L., D. E. Knuth, & O. Patashnik (1994) Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.
6. Merris, R. (2000) The p-Stirling numbers, Turkish J. Math. , 24, 379–399. 79
7. Mihoubi, M. & M. Rahmani, The partial r-Bell polynomials, arXiv:1308.0863.
8. Nyul, G. & G. R´acz (2014) The r-Lah numbers, Discrete Math., http://dx.doi.org/10.1016/j.disc.2014.03.029 (in press).
9. Stanley, R. P. (1997) Enumerative Combinatorics, Vol. I, Cambridge University Press.
10. Wagner, C. (1996) Generalized Stirling and Lah numbers, Discrete Math., 160, 199–218.