David-Barton identities and Stirling permutations

In this talk, we present a survey on the study of David-Barton identities and Stirling permutations. By using an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials, we find that for any gamma-positive polynomial, there exists a David-Barton type identity. We then discuss some enumerative polynomials on Stirling permutations, including ascent-plateau polynomials, left ascent-plateau polynomials and flag ascent-plateau polynomials.