Abstract
After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones.
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Benoumhani, M.: On Whitney numbers of Dowling lattices. Discrete Math. 159, 13–33 (1996)
Benoumhani, M.: On some numbers related to Whitney numbers of Dowling lattices. Adv. Appl. Math. 19, 106–116 (1997)
Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)
Carlitz, L.: Weighted Stirling numbers of the first and second kind I. Fibonacci Q. 18, 147–162 (1980)
Cheon, G.-S., Jung, J.-H.: \(r\)-Whitney numbers of Dowling lattices. Discrete Math. 312, 2337–2348 (2012)
Corcino, R.B., Corcino, C.B.: On generalized Bell polynomials. Discrete Dyn. Nat. Soc., Article 623456 (2011)
Corcino, R.B., Corcino, C.B., Aldema, R.: Asymptotic normality of the \((r,\beta )\)-Stirling numbers. Ars Combin. 81, 81–96 (2006)
Davenport, H., Pólya, G.: On the product of two power series. Can. J. Math. 1, 1–5 (1949)
Dowling, T.A.: A class of geometric lattices based on finite groups. J. Comb. Theory Ser. B 14, 61–86 (1973); erratum, 15, 211 (1973)
Gyimesi, E., Nyul, G.: New combinatorial interpretations of \(r\)-Whitney and \(r\)-Whitney-Lah numbers. Submitted
Hsu, L.C., Shiue, P.J.-S.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20, 366–384 (1988)
Liu, L.L., Wang, Y.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453–476 (2007)
Mangontarum, M.M., Macodi-Ringia, A. P., Abdulcarim, N. S.: The translated Dowling polynomials and numbers. Int. Sch. Res. Not., Article 678408 (2014)
Merris, R.: The \(p\)-Stirling numbers. Turkish J. Math. 24, 379–399 (2000)
Mező, I.: A new formula for the Bernoulli polynomials. Results Math. 58, 329–335 (2010)
Mező, I.: The \(r\)-Bell numbers, J. Integer. Seq. 14, Article 11.1.1 (2011)
Mező, I.: A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices. Discrete Math. 328, 88–95 (2014)
Mihoubi, M., Rahmani, M.: The partial \(r\)-Bell polynomials. arXiv:1308.0863
Rahmani, M.: Some results on Whitney numbers of Dowling lattices. Arab J. Math. Sci. 20, 11–27 (2014)
Remmel, J.B., Wachs, M.L.: Rook theory, generalized Stirling numbers and \((p, q)\)-analogues. Electron. J. Comb. 11/1, R84 (2004)
Wang, D.G.L.: On colored set partitions of type \(B_{n}\). Cent. Eur. J. Math. 12, 1372–1381 (2014)
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Research was supported by Grant 115479 from the Hungarian Scientific Research Fund, and by the ÚNKP-17-4 New National Excellence Program of the Ministry of Human Capacities.
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Gyimesi, E., Nyul, G. A comprehensive study of \({\varvec{r}}\)-Dowling polynomials. Aequat. Math. 92, 515–527 (2018). https://doi.org/10.1007/s00010-017-0538-z
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DOI: https://doi.org/10.1007/s00010-017-0538-z