  In this paper, we characterize a duality relation between Eulerian
recurrences and Eulerian recurrence systems, which generalizes and unifies
Hermite-Biehler decompositions of several enumerative polynomials, including
flag descent polynomials for hyperoctahedral group, flag ascent-plateau
polynomials for Stirling permutations, up-down run polynomials for symmetric
group and alternating run polynomials for hyperoctahedral group. As
applications, we derive some properties of associated enumerative polynomials.
In particular, we find that both the ascent-plateau polynomials and left
ascent-plateau polynomials for Stirling permutations are alternatingly
increasing, and so they are unimodal with modes in the middle.
