A new approach to the r-Whitney numbers by using combinatorial differential calculus

[1] M. Benoumhani, On some numbers related to Whitney numbers of Dowling lattices, Adv. Appl. Math., 19 (1997), 106–116.10.1006/aama.1997.0529Search in Google Scholar

[2] M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math., 159 (1996), 13–33.10.1016/0012-365X(95)00095-ESearch in Google Scholar

[3] F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-Like Structures, Encyclopedia of Mathematic and its Applications, Volume 67, Cambridge University Press, Cambridge, 1998.10.1017/CBO9781107325913Search in Google Scholar

[4] A. Z. Broder, The r-Stirling numbers, Discrete Math., 49 (1984), 241–259.10.1016/0012-365X(84)90161-4Search in Google Scholar

[5] D. Callan, S.-M. Ma, T. Mansour, Some combinatorial arrays related to the Lotka-Volterra system, Electron. J. Combin., 22 (2) (2015), # P2.22.10.37236/4413Search in Google Scholar

[6] W. Y. C. Chen. Context-Free grammars, differential operators and formal power series, Theoret. Comput. Sci., 117 (1993), 113–129.10.1016/0304-3975(93)90307-FSearch in Google Scholar

[7] W. Y. C. Chen, A. M. Fu, Context-free grammars for permutations and increasing trees, Adv. Appl. Math., 82 (2017), 58–82.10.1016/j.aam.2016.07.003Search in Google Scholar

[8] G.-S. Cheon, J.-H. Jung, r-Whitney numbers of Dowling lattices, Discrete Math., 312 (15) (2012), 2337–2348.10.1016/j.disc.2012.04.001Search in Google Scholar

[9] C. B. Corcino, R. B. Corcino, R. J. Gasparin, Equivalent asymptotic formulas of second kind r-Whitney numbers, Integral Transforms Spec. Funct., 26(3)(2015), 192–202.10.1080/10652469.2014.979410Search in Google Scholar

[10] C. B. Corcino, R. B. Corcino, I. Mező, J. L. Ramírez, Some polynomials associated with the r-Whitney numbers, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), 1–25.10.1007/s12044-018-0406-3Search in Google Scholar

[11] T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B, 14 (1973) 61–86. erratum; J. Combin. Theory Ser. B 15 (1973), 211.Search in Google Scholar

[12] E. Gyimesi, G. Nyul, A comprehensive study of r-Dowling polynomials, Aequationes Math., 92 (3) (2018), 515–527.10.1007/s00010-017-0538-zSearch in Google Scholar

[13] R. Hao, L. Wang, H. Yang, Context-free grammars for triangular arrays, Acta Math. Sin. (Engl. Ser.), 31 (3) (2015), 445–455.10.1007/s10114-015-4209-5Search in Google Scholar

[14] O. Herscovici, T. Mansour, Identities involving Touchard polynomials derived from umbral calculus, Adv. Stud. Contemp. Math., (Kyungshang) 25 (1) (2015), 39–46.Search in Google Scholar

[15] A. Joyal, Une théorie combinatoire des séries formelles, Adv. Math., 42 (1981), 1–82.10.1016/0001-8708(81)90052-9Search in Google Scholar

[16] P. Leroux, G. X. Viennot, Combinatorial resolution of systems of differential equations, I. Ordinary differential equations. In Combinatoire Enumérativé (1986), G. Labelle, P. Leroux, Eds., no. 1234 in Lecture Notes in Mathematics, Springer-Verlag, 210–245.10.1007/BFb0072518Search in Google Scholar

[17] S.-M. Ma, Some combinatorial arrays generated by context-free grammars, European J. Combin., 34 (2013), 1081–1091.10.1016/j.ejc.2013.03.002Search in Google Scholar

[18] T. Mansour, J. L. Ramírez, M. Shattuck, A generalization of the r-Whitney numbers of the second kind, J. Comb., 8 (1) (2017) 29–55.10.4310/JOC.2017.v8.n1.a2Search in Google Scholar

[19] M. Méndez, Combinatorial differential operators in: Faá di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees. ArXiv preprint, available online at http://arxiv.org/pdf/1610.03602v1.pdfSearch in Google Scholar

[20] M. Merca. A convolution for the complete and elementary symmetric functions, Aequationes Math., 86 (3) (2013), 217–229.10.1007/s00010-012-0170-xSearch in Google Scholar

[21] M. Merca, A note on the r-Whitney numbers of Dowling lattices, C. R. Math. Acad. Sci. Paris., 351 (16-17) (2013), 649–655.10.1016/j.crma.2013.09.011Search in Google Scholar

[22] D. Merlini, D. G. Rogers, R. Sprugnoli, M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canadian J. Math., 49 (1997), 301–320.10.4153/CJM-1997-015-xSearch in Google Scholar

[23] D. Merlini, R. Sprugnoli, M. C. Verri, The Cauchy numbers, Discrete Math., 306 (2006), 1906–1920.10.1016/j.disc.2006.03.065Search in Google Scholar

[24] I. Mező, A new formula for the Bernoulli polynomials, Result. Math., 58 (3) (2010), 329–335.10.1007/s00025-010-0039-zSearch in Google Scholar

[25] I. Mező, J. L. Ramírez, The linear algebra of the r-Whitney matrices, Integral Transforms Spec. Funct., 26 (3) (2015), 213–225.10.1080/10652469.2014.984180Search in Google Scholar

[26] I. Mező, J. L. Ramírez, Some identities of the r-Whitney numbers, Aequationes Math., 90 (2) (2016), 393–406.10.1007/s00010-015-0404-9Search in Google Scholar

[27] M. Mihoubi, M. Rahmani, The partial r-Bell polynomials, Afr. Mat., 28 (2017), 1167–1183.10.1007/s13370-017-0510-zSearch in Google Scholar

[28] M. Mihoubi, M. Tiachachat, Some applications of the r-Whitney numbers, C. R. Math. Acad. Sci. Paris., 352 (12) (2014), 965–969.10.1016/j.crma.2014.08.001Search in Google Scholar

[29] J. L. Ramírez, M. Shattuck, Generalized r-Whitney numbers of the first kind, Ann. Math. Inform., 46 (2016), 175–193.Search in Google Scholar

[30] D. G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math., 22 (1978), 301–310.10.1016/0012-365X(78)90063-8Search in Google Scholar

[31] S. Roman, The Umbral Calculus, Pure and Applied Mathematics, 111. Academic Press Inc. (1984).Search in Google Scholar

[32] S. Roman, G.-C. Rota,The Umbral Calculus, Adv. Math., 27 (1978), 95–188.10.1016/0001-8708(78)90087-7Search in Google Scholar

[33] L. W. Shapiro, S. Getu, W. Woan, L. Woodson, The Riordan group, Discrete Appl. Math., 34 (1991), 229–239.10.1016/0166-218X(91)90088-ESearch in Google Scholar

[34] M. Z. Spivey, A generalized recurrence for Bell numbers, J. Integer Seq., 11 (2008), Article 08.2.5.Search in Google Scholar

[35] W. Wang, T. Wang, Generalized Riordan arrays, Discrete Math., 308 (2008), 6466–6500.10.1016/j.disc.2007.12.037Search in Google Scholar

[36] A. Xu, Extensions of Spivey’s Bell number formula, Electron. J. Combin., 19 (2), #P6(2012).10.37236/2146Search in Google Scholar

[37] A. Xu, T. Zhou, Some identities related to the r-Whitney numbers, Integral Transforms Spec. Funct., 27 (11) (2016), 920–929.10.1080/10652469.2016.1229316Search in Google Scholar