Jesús Salas

Universidad Carlos III de Madrid

"Bivariate generating functions for a class of linear recurrences in Combinatorics: Structure and Applications"

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.