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

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.