Lah number

Abstract: Preface. 1. Basic Combinatorics and Probability. 2. Stirling Numbers. 3. Generalized Stirlized and LAH Numbers. 4. Occupancy Distributions. 5. Sequential Occupancy Distributions. 6. Convolutions of Truncated Distributions. 7. Compound and Mixture Distributions. Appendix: Hints and Answers to Exercises. References. Author Index. Subject Index.

Abstract: Sequences of generalized Stirling numbers of both kinds are introduced. These sequences of triangles (i.e. infinite-dimensional lower triangular matrices) of numbers will be denoted by S2(k;n,m) and S1(k;n,m) with k ∈ Z. The original Stirling number triangles of the second and first kind arise when k = 1. S2(2;n,m) is identical with the unsigned S1(2;n,m) triangle, called S1p(2;n,m), which also represents the triangle of signless Lah numbers. Certain associated number triangles, denoted by s2(k;n,m) and s1(k;n,m), are also defined. Both s2(2;n,m) and s1(2;n + 1,m + 1) form Pascal’s triangle, and s2(−1, n,m) turns out to be Catalan’s triangle. Generating functions are given for the columns of these triangles. Each S2(k) and S1(k) matrix is an example of a Jabotinsky matrix. The generating functions for the rows of these triangular arrays therefore constitute exponential convolution polynomials. The sequences of the row sums of these triangles are also considered. These triangles are related to the problem of obtaining finite transformations from infinitesimal ones generated by x d dx , for k ∈ Z. AMS MSC numbers: 11B37, 11B68, 11B83, 11C08, 15A36 1 Overview Stirling’s numbers of the second kind (also called subset numbers), and denoted by S2(n,m) (or {

Abstract: Let the numbers P(r,n,k) be defined by P(r,n,k):=P"r(H"n^(^1^)-H"k^(^1^),...,H"n^(^r^)-H"k^(^r^)), where P"r(x"1,...,x"r)=(-1)^rY"r(-0!x"1,-1!x"2,...,-(r-1)!x"r) and Y"r are the exponential complete Bell polynomials. By observing that the numbers P(r,n,k) generate two Riordan arrays, we establish several general summation formulas, from which series of harmonic number identities are obtained. In particular, some of these harmonic number identities also involve other special combinatorial sequences, such as the Stirling numbers of both kinds, the Lah numbers, the Bernoulli numbers and polynomials and the Cauchy numbers of both kinds.