Mark Shattuck

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 2, Pages 145-154

DOI: 10.7546/nntdm.2019.25.2.145-154

**Full paper (PDF, 201 Kb)**

## Details

### Authors and affiliations

Mark Shattuck

*Institute for Computational Science & Faculty of Mathematics and Statistics
Ton Duc Thang University, Ho Chi Minh City, Vietnam
*

### Abstract

Recently, three new Bell number formulas were proven using algebraic methods, one of which extended an earlier identity of Gould–Quaintance and another a previous identity of Spivey. Here, making use of combinatorial arguments to establish our results, we find generalizations of these formulas in terms of the *r*-Dowling polynomials. In two cases, weights of the form *a ^{i}* and

*b*may be replaced by arbitrary sequences of variables

^{j}*x*and

_{i}*y*which yields further generalizations. Finally, a second extension of one of the formulas is found that involves generalized Stirling polynomials and leads to analogues of this formula for other counting sequences.

_{j}### Keywords

- Bell numbers
*r*-Dowling polynomials*r-*Whitney numbers- Polynomial generalization

### 2010 Mathematics Subject Classification

- 05A19
- 11B73

### References

- Benoumhani, M. (1997). On some numbers related to Whitney numbers of Dowling lattices, Adv. in Appl. Math., 19, 106–116.
- Cheon, G.-S. & Jung, J.-H. (2012).
*r*-Whitney numbers of Dowling lattices, Discrete Math., 312, 2337–2348. - Dowling, T. A. (1973). A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B, 14, 61–86. (Erratum: (1973). J. Combin. Theory Ser. B, 15, 211.)
- Gould, H. W. & Quaintance, J. (2008). Implications of Spivey’s Bell number formula, J.Integer Seq., 11, Art. 08.3.7.
- Komatsu, T. & Pita-Ruiz, C. (2018). Some formulas for Bell numbers,Filomat, 32(11), 3881–3889.
- Mansour, T., Ramirez, J. L. & Shattuck, M. (2017). A generalization of the
*r*-Whitney numbers of the second kind,J. Comb., 8(1), 29–55. - Mansour, T., Schork, M. & Shattuck, M. (2011). On a new family of generalized Stirling and Bell numbers, Electron. J. Combin., 18, #P77.
- Mezo, I. (2010). A new formula for the Bernoulli polynomials, Results Math., 58(3), 329–335.
- Mezo, I. (2012). The dual of Spivey’s Bell number formula, J. Integer Seq., 15, Art. 12.2.4.
- Mihoubi, M. & Belbachir, H. (2014). Linear recurrences for
*r*-Bell polynomials, J. Integer Seq., 17, Art. 14.10.6. - Rahmani, M. (2014). Some results on Whitney numbers of Dowling lattices,Arab J. Math. Sci., 20(1), 11–27.
- Shattuck, M. (2016). Generalizations of Bell number formulas of Spivey and Mezo, Filomat,30:10, 2683–2694.
- Sloane, N. J. A. (2010).On-line Encyclopedia of Integer Sequences, Available online at: http://oeis.org.
- Spivey, M. Z. (2008). A generalized recurrence for Bell numbers, J. Integer Seq., 11, Art.08.2.5.
- Xu, A. (2012). Extensions of Spivey’s Bell number formula, Electron. J. Combin., 19(2),#P6.

## Related papers

## Cite this paper

Shattuck, M.(2019). Some combinatorial identities for the *r*-Dowling polynomials. *Notes on Number Theory and Discrete Mathematics*, 25(2), 145-154, DOI: 10.7546/nntdm.2019.25.2.145-154.