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In this paper, we focus on a "local property" of permutations: value-peak. A permutation $\sigma$ has a value-peak $\sigma(i)$ if $\sigma(i-1) < \sigma(i)>\sigma(i+1)$ for some $i\in[2,n-1]$. Define $VP(\sigma)$ as the set of value-peaks of the permutation $\sigma$. For any $S\subseteq [3,n]$, define $VP_n(S)$ such that $VP(\sigma)=S$. Let ${\cal P}_n=\{S\mid VP_n(S)\neq\emptyset\}$. we make the set ${\cal P}_n$ into a poset $\mathfrak{ P}$$_n$ by defining $S\preceq T$ if $S\subseteq T$ as sets. We prove that the poset $\mathfrak{ P}$$_n$ is a simplicial complex on the set $[3,n]$ and study some of its properties. We give enumerative formulae of permutations in the set $VP_n(S)$.

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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.