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Circular Peaks and Hilbert Series (2008)
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@MISC{Bouchard08circularpeaks,
author = {Pierre Bouchard and Jun Ma and Yeong-nan Yeh},
title = {Circular Peaks and Hilbert Series},
year = {2008}
}
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Abstract
The circular peak set of a permutation σ is the set {σ(i) | σ(i−1) < σ(i)> σ(i+1)}. Let Pn be the set of all the subset S ⊆ [n] such that there exists a permutation σ which has the circular set S. We can make the set Pn into a poset Pn by defining S ≼ T if S ⊆ T as sets. In this paper, we prove that the poset Pn is a simplicial complex on the vertex set [3,n]. We study the f-vector, the f-polynomial, the reduced Euler characteristic, the Möbius function, the h-vector and the h-polynomial of Pn. We also derive the zeta polynomial of Pn and give the formula for the number of the chains in Pn. By the poset Pn, we define two algebras APn and BPn. We consider the Hilbert polynomials and the Hilbert series of the algebra APn and BPn
