Generalization of Stirling Number of the Second Kind and Combinatorial Identity

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: ∑AC11 AC22 ···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).