Eulerian pairs and Eulerian recurrence systems

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between recurrences recurrence systems. This generalizes unifies decompositions several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating hyperoctahedral flag descent groups ascent-plateau Stirling permutations. derive some properties associated polynomials. particular, prove alternatingly increasing property interlacing left