  In the context of Stirling polynomials, Gessel and Stanley introduced the
definition of Stirling permutation, which has attracted extensive attention
over the past decades. Recently, we introduced Stirling permutation code and
provided numerous equidistribution results as applications. The purpose of the
present work is to further analyse Stirling permutation code. First, we derive
an expansion formula expressing the joint distribution of the types $A$ and $B$
descent statistics over the hyperoctahedral group, and we also find an
interlacing property involving the zeros of its coefficient polynomials. Next,
we prove a strong connection between signed permutations in the hyperoctahedral
group and Stirling permutations. Furthermore, we investigate unified
generalizations of the trivariate second-order Eulerian polynomials and
ascent-plateau polynomials. Using Stirling permutation codes, we provide
expansion formulas for eight-variable and seventeen-variable polynomials, which
imply several $e$-positive expansions and clarify the connections among several
statistics. Our results generalize the results of B\'ona, Chen-Fu, Dumont,
Janson, Haglund-Visontai and Petersen.
