On the h-triangles of sequentially (Sr) simplicial complexes via algebraic shifting

Abstract

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S_r) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S_r). Let Δ be a (d−1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h_{i, j}(Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h_{i, j}(Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S_r) and for every i and j with 0≤j≤i≤r−1, the equality h_{i, j}(Δ)=h_{i, j}(Γ(Δ)) holds true.