Circular Peaks and Hilbert Series

The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid \sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the subset $S\subseteq [n]$ such that there exists a permutation $\sigma$ which has the circular set $S$. We can make the set $\mathcal{P}_n$ into a poset $\mathscr{P}_n$ by defining $S\preceq T$ if $S\subseteq T$ as sets. In this paper, we prove that the poset $\mathscr{P}_n$ is a simplicial complex on the vertex set $[3,n]$. We study the $f$-vector, the $f$-polynomial, the reduced Euler characteristic, the M$\ddot{o}$bius function, the $h$-vector and the $h$-polynomial of $\mathscr{P}_n$. We also derive the zeta polynomial of $\mathscr{P}_n$ and give the formula for the number of the chains in $\mathscr{P}_n$. By the poset $\mathscr{P}_n$, we define two algebras $\mathcal{A}_{\mathscr{P}_n}$ and $\mathcal{B}_{\mathscr{P}_n}$. We consider the Hilbert polynomials and the Hilbert series of the algebra $\mathcal{A}_{\mathscr{P}_n}$ and $\mathcal{B}_{\mathscr{P}_n}$.