Value-Peaks of Permutations

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 $VP_n(S)$.