| 释义 |
Dobinski's FormulaGives the  th Bell Number,
 | (1) |
 , yielding
 | (2) |
 | (3) |
 | (4) |
 gives the identity (Dobinski 1877; Rota 1964; Berge 1971, p. 44; Comtet 1974, p. 211; Roman1984, p. 66; Lupas 1988; Wilf 1990, p. 106; Chen and Yeh 1994; Pitman 1997).
References
Berge, C. Principles of Combinatorics. New York: Academic Press, 1971.Chen, B. and Yeh, Y.-N. ``Some Explanations of Dobinski's Formula.'' Studies Appl. Math. 92, 191-199, 1994. Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Boston, MA: Reidel, 1974. Dobinski, G. ``Summierung der Reihe  für  , 2, 3, 4, 5, ....'' Grunert Archiv (Arch. Math. Phys.) 61, 333-336, 1877. Foata, D. La série génératrice exponentielle dans les problèmes d'énumération. Vol. 54 of Séminaire de Mathématiques supérieures. Montréal, Canada: Presses de l'Université de Montréal, 1974. Lupas, A. ``Dobinski-Type Formula for Binomial Polynomials.'' Stud. Univ. Babes-Bolyai Math. 33, 30-44, 1988. Pitman, J. ``Some Probabilistic Aspects of Set Partitions.'' Amer. Math. Monthly 104, 201-209, 1997. Roman, S. The Umbral Calculus. New York: Academic Press, 1984. Rota, G.-C. ``The Number of Partitions of a Set.'' Amer. Math. Monthly 71, 498-504, 1964. Wilf, H. Generatingfunctionology, 2nd ed. San Diego, CA: Academic Press, 1990. |