Circular Peaks and Hilbert Series

2008/06/03 by Pierre Bouchard, Bouchard, Pierre, Jun Ma +4
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.0806.0434

openalex publication_date 2008/06/03 · arxiv created 2008/06/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The circular peak set of a permutation σ is the set \σ(i)| σ(i-1)<σ(i)>σ(i+1)\. Let P_n 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 P_n into a poset \mathscrP_n by defining S\preceq T if S⊆ T as sets. In this paper, we prove that the poset \mathscrP_n is a simplicial complex on the vertex set [3,n]. We study the f-vector, the f-polynomial, the reduced Euler characteristic, the Mobius function, the h-vector and the h-polynomial of \mathscrP_n. We also derive the zeta polynomial of \mathscrP_n and give the formula for the number of the chains in \mathscrP_n. By the poset \mathscrP_n, we define two algebras A_\mathscrP_n and B_\mathscrP_n. We consider the Hilbert polynomials and the Hilbert series of the algebra A_\mathscrP_n and B_\mathscrP_n.

Related

• An order on circular permutations
• Enumerations for Permutations by Circular Peak Sets
• Hilbert scheme of points on cyclic quotient singularities of type (p,1)
• Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings
• Sets of Hilbert series and their applications