David-Barton type identities and alternating run polynomials

Abstract

In this paper, we first consider an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials. By using this alternate formulation, we see that for any γ-positive polynomial, there exists a David-Barton type identity. We then consider the joint distribution of cycle runs and cycles over the set of permutations. Furthermore, we introduce the definition of semi-γ-positive polynomial. The γ-positivity of a polynomial f(x) is a sufficient (not necessary) condition for the semi-γ-positivity of f(x). We show that the alternating run polynomial of dual Stirling permutations is semi-γ-positive but not γ-positive.