  In this paper, we first consider a generalization of the David-Barton
identity which relate the alternating run polynomials to Eulerian polynomials.
By using context-free grammars, we then present a combinatorial interpretation
of a family of q-alternating run polynomials. Furthermore, we introduce the
definition of semi-gamma-positive polynomial and we show the
semi-gamma-positivity of the alternating run polynomials of dual Stirling
permutations. A connection between the up-down run polynomials of permutations
and the alternating run polynomials of dual Stirling permutations is
established.
