Stirling numbers 1 Stirling numbers of the second kind The Stirling numbers S(m,n) of the second kind count the number of ways to partition an m-element set into n equivalence classes. As a consequence, the number of functions from an m-element set onto an n-element set (with distinct elements on both sides) is S(m,n)·n! Oct 20, 2017 · There are two ways of calculating Stirling numbers of the second kind. First,they can be calculated recursively; i.e, with reference to lower order Stirling numbers of the second kind. S(m,n) = S(m – 1,n – 1) + nS(m – 1,n). Before we define the Stirling numbers of the first kind, we need to revisit permutations. As we mentioned in section 1.7, we may think of a permutation of $[n]$ either as a reordering of $[n]$ or as a bijection $\sigma\colon [n]\to[n]$.

formula, the Nörlund polynomials [generalized Bernoulli numbers], and the Stirling numbers and binomial coefficients with negative indices [duality property]. The application of (6) in (10) and (11) gives the Sun’s identities [23, 32] for Stirling numbers of the first and second kind. References