Authors and affiliations
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.
- Stirling numbers
- r-Whitney numbers
- q-Stirling numbers
- Benjamin, A. T. & J. J. Quinn (2003) Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America.
- Broder, A. Z. (1984) The r-Stirling numbers, Discrete Math., 49, 241–259.
- Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., 15, 987–1000.
- Cheon, G.-S. & J.-H. Jung (2012) r-Whitney numbers of Dowling lattices, Discrete Math., 312, 2337–2348.
- Graham, R. L., D. E. Knuth, & O. Patashnik (1994) Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.
- Merris, R. (2000) The p-Stirling numbers, Turkish J. Math. , 24, 379–399. 79
- Mihoubi, M. & M. Rahmani, The partial r-Bell polynomials, arXiv:1308.0863.
- 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).
- Stanley, R. P. (1997) Enumerative Combinatorics, Vol. I, Cambridge University Press.
- Wagner, C. (1996) Generalized Stirling and Lah numbers, Discrete Math., 160, 199–218.
Cite this paperAPA
Shattuck, M. (2015). A generalized recurrence formula for Stirling numbers and related sequences. Notes on Number Theory and Discrete Mathematics, 21(4), 74-80.Chicago
Shattuck, Mark. “A Generalized Recurrence Formula for Stirling Numbers and Related Sequences.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 74-80.MLA
Shattuck, Mark. “A Generalized Recurrence Formula for Stirling Numbers and Related Sequences.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 74-80. Print.