A Multitude of Expressions for the Stirling Numbers of the First Kind

It is shown that the Stirling numbers of the first kind can be expressed in the form [n k ] = ∑n−1 j1<j2<···<jk−1 (n−1)! αQ[j1,j2,...,jk−1] , where Q is a product of k − 1 linear factors in the indices j1, j2, . . . , jk−1 and α is a normalization coefficient determined by the condition [ k k ] = 1. Several types of Q’s are shown to yield Stirling numbers (“be Stirling”), and some more are conjectured to do so. The complete characterization of the set of Q’s that are Stirling is not yet available. This set can be divided into subsets, within each of which different Q’s are related by permutational symmetries. The case Q = j1 · j2 · · · jk−1 is due to Adamchik (1997).