Stirling Numbers and Generalizations

Abstract

Abstract The Stirling numbers of the first and second kind enumerate the permutations of a finite set that are decomposed into a given number of cycles and the partitions of a finite set into a given number of subsets, respectively. Basic properties of these numbers and their generalizations of probabilistic and statistical interest are reviewed. Specifically, explicit expressions, generating functions, recurrence relations, and other useful properties of these numbers are presented. Further, the Lah numbers and the generalized factorial coefficients, closely related to the Stirling numbers, are discussed. The noncentral and the associated Stirling and related numbers are briefly presented. The generalized Stirling and Lah numbers and, in particular, the q ‐Stirling and q ‐Lah numbers are only quoted. Probabilistic and statistical applications of the Stirling and related numbers in expressing the probability mass function of certain (a) urn‐model distributions, (b) occupancy distributions, (c) convolutions of truncated discrete distributions and (d) compound discrete distributions are exemplified.