Some results on Whitney numbers of Dowling lattices. (English) Zbl 1377.11032
Summary: In this paper, we study some properties of Whitney numbers of Dowling lattices and related polynomials. We answer the following question: there is a relation between Stirling and Eulerian polynomials. Can we find a new relation between Dowling polynomials and other polynomials generalizing Eulerian polynomials? In addition, some congruences for the Dowling numbers are given.
MSC:
| 11B73 | Bell and Stirling numbers |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 05A15 | Exact enumeration problems, generating functions |
Keywords:
Bell polynomials; congruences; Dowling lattices; Eulerian polynomials; Hankel determinant; Whitney numbersReferences:
| [1] | Belbachir, H.; Mihoubi, M., A generalized recurrence for Bell polynomials: an alternate approach to Spivey and Gould-Quaintance formulas, European J. Combin., 30, 1254-1256 (2009) · Zbl 1187.05007 |
| [2] | Benoumhani, M., On Whitney numbers of Dowling lattices, Discrete Math., 159, 13-33 (1996) · Zbl 0861.05004 |
| [3] | Benoumhani, M., On some numbers related to Whitney numbers of Dowling lattices, Adv. Appl. Math., 19, 106-116 (1997) · Zbl 0876.05001 |
| [4] | Benoumhani, M., Log-concavity of Whitney numbers of Dowling lattices, Adv. Appl. Math., 22, 186-189 (1999) · Zbl 0918.05003 |
| [5] | Boyadzhiev, K. N., The Euler series transformation and the binomial identities of Ljunggren, Munarini and Simons, Integers, 10, 265-271 (2010) · Zbl 1226.05015 |
| [6] | Broder, A. Z., The \(r\)-Stirling numbers, Discrete Math., 49, 3, 241-259 (1984) · Zbl 0535.05006 |
| [7] | Chen, K. W., Identities from the binomial transform, J. Number Theory, 124, 142-150 (2007) · Zbl 1120.05007 |
| [8] | Cheon, G.-S.; Jung, J.-H., The \(r\)-Whitney numbers of Dowling lattices, Discrete Math., 312, 15, 2337-2348 (2012) · Zbl 1246.05009 |
| [9] | Comtet, L., Advanced Combinatorics (1974), D. Reidel Publishing Co. · Zbl 0283.05001 |
| [10] | Dowling, T. A., A class of gemometric lattices bases on finite groups, J. Combin. Theory Ser. B, 14, 61-86 (1973), Erratum J. Combin. Theory Ser. B 15 (1973) 211 · Zbl 0247.05019 |
| [11] | Frobenius, G., Über die Bernoullischen und die Eulerschen Polynome, Sitzungsberichte der Preussische Akademie der Wissenschaften, 809-847 (1910) · Zbl 1264.11013 |
| [12] | Gessel, I., Congruences for Bell and Tangent numbers, Fibonacci Quart., 19, 2, 137-144 (1981) · Zbl 0451.10008 |
| [13] | Layman, J. W., The Hankel transform and some of its properties, J. Integer Seq., 4 (2001), Article 01.1.5 · Zbl 0978.15022 |
| [14] | Mansour, T.; Shattuck, M., A recurrence related to the Bell numbers, Integers, 11, A67 (2011) · Zbl 1264.11014 |
| [15] | Mező, I., A new formula for the Bernoulli polynomials, Results Math., 58, 329-335 (2010) · Zbl 1237.11010 |
| [16] | Mező, I., The r-Bell numbers, J. Integer Seq., 14 (2011), Article 11.1.1 · Zbl 1205.05017 |
| [17] | Radoux, C., Calcul effectif de certains déterminants de Hankel, Bull. Soc. Math. Belg. Sér. B, 31, 1, 49-55 (1979) · Zbl 0439.10006 |
| [18] | Rahmani, M., The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algebra Appl., 438, 219-230 (2013) · Zbl 1257.05010 |
| [19] | Riordan, J., An Introduction to Combinatorial Analysis (2002), Dover Publications Inc. |
| [20] | Suter, R., Two analogues of a classical sequence, J. Integer Seq., 3 (2000), Article 00.1.8 · Zbl 1004.05002 |
| [21] | Tanny, S. M., On some numbers related to the Bell numbers, Can. Math. Bull., 17, 733-738 (1975) · Zbl 0304.10007 |
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