Universidad Carlos III de Madrid

In this talk, I will discuss a complete solution in terms of exponential
generating functions for the "research problem" 6.94 posed in 1989 by
Grahan, Knuth, and Patashnik in their book Concrete Mathematics.
This problem is related to a general family of
two-parameter linear recurrences that contains many of the best-known
sequences of Enumerative Combinatorics including binomial coefficients, and
Stirling, Ward, Lah, and Eulerian numbers. One interesting and curious
feature of these recurrences is that there are some families giving rise
to exactly the same combinatorial numbers. As an application, I will
discuss a natural three-parameter generalization of the Eulerian and Ward
numbers, such that they form Riordan inverse pairs. For the former ones,
I will describe their combinatorial interpretation in terms of generalized
Stirling permutations.