The Stirling numbers of second kind and related problems are widely used in combinatorial mathematics and number theory, and there are a lot of research results. This article discuss the function:
∑AC11AC22 ···ACkk (C1 C2 ··· Ck=N-K, Ci≥0), obtain its calculation formula and a series of conclusions, which generalize the results of existing literature, and further obtain the combinatorial identity: ∑(-1)K-i*C(K-1,K-i)C(A-1 i,N-1)=C(A,N-K).
References
[1]
Hardy, G.H. (2010) An Introduction to the Theory of Numbers. 6th Edition. Zhang, F. Translation. People’s Post and Telecommunications Press, Beijing.
[2]
Li, C.X. (1995) The Expression of Stirling Number of the Second Kind and Others. Journal of Hubei Normal University: Philosophy and Social Sciences, 6, 72-76.
[3]
Huang, R.H. and Chen, B.E. (2011) Some Conclusions on the Generalized Stirling Number of the Second Kind. Journal of Hanshan Normal University, 3, 29-31.
[4]
Zhang, F.L. (2011) An Identity of Stirling Numbers of the Second Kind. Journal of Weinan Teachers College, 12, 14-16.
[5]
Editorial Board of Handbook of Modern Applied Mathematics (2002) Handbook of Modern Applied Mathematics Discrete Mathematics Volume. Tsinghua University Press, Beijing.