Stirling permutation codes. II

Abstract: 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.