Generalizations of the Stirling, Lah, and cycle numbers of the first and second kinds

As Stirling, Lah, and cycle numbers of the second kind satisfy very similar recurrence relations, one might expect that these numbers can be derived from one recurrence relation. This is indeed true, and this one recurrence provides one of several ways to define the SLC numbers of the second kind, which generalize the Stirling, Lah, and cycle numbers of the second kind. In the first chapter, five equivalent ways to define SLC numbers of the first and second kind are presented, and relationships between the numbers of the first and second kind are explored. In the second chapter, special cases of the SLC numbers of the first and second kind are considered. In particular, we find that binomial and q-binomial coefficients are SLC numbers, and that the SLC numbers provide a generalization of Louis Comtet's generalization of Stirling numbers of the first and second kind.