Generalized Stirling and Lah numbers

The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces, and chains. Within this theory three generalizations of Stifling numbers of the second kind, and of Lah numbers, are developed. 1. Stirling numbers and their formal generalizations The nota t ional convent ions of this paper are as follows: N = {0,1,2 . . . . }, P = {1,2,. . . }, [0] = 0 , and I-n] = {1 . . . . . n} for n~ P. Empty sums take the value 0 and empty products the value 1. Also, x ° = x -° = x ° = 1 for all x (including x = 0), and f o r n ~ P , x ~ = x ( x 1 ) . . . ( x n + 1) a n d x n = x ( x + 1 ) . ( x + n 1 ) . As enumera to r of part i t ions of I-n] with k blocks, the Stirling number of the second kind S(n, k) plays a central role in elementary combinatorics . No t surprisingly, apar t from the boundary values S(n,O) = J,.o and S(n,k) = 0 for 0 ~< n < k, there are many representations of these numbers. F r o m the s tandpoint of generalizations pursued in this paper these representations fall natural ly into three classes: Class I 1 ~+ n! = m m S(n, k) k! n~+ .-n k = n n l ! nk! S(n,k) = ~ . ~ S(n j , k 1), (1.1)