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Circular Peaks and Hilbert Series | | Pierre Bouchard
; Jun Ma
; Yeong-Nan Yeh
; | | Date: |
3 Jun 2008 | | Abstract: | 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 $Ssubseteq [n]$ 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 $Spreceq T$ if $Ssubseteq T$ 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 $mathcal{B}_{mathscr{P}_n}$. | | Source: | arXiv, 0806.0434 | | Services: | Forum | Review | PDF | Favorites | |
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