Recently, Haghighi, Terai, Yassemi, and ZaareNahandi introduced the notion
of a sequentially (S_{r}) simplicial complex. This notion gives a
generalization of two properties for simplicial complexes: being
sequentially CohenMacaulay and satisfying Serre's condition (S_{r}). Let
∆ be a (d−1)dimensional simplicial complex with Γ(∆)
as its algebraic shifting. Also let (h_{i,j}(∆))_{0 ≤ j ≤ i ≤ d} be the htriangle of ∆ and (h_{i,j}(Γ(∆)))_{0 ≤ j ≤ i ≤ d} be the htriangle of Γ(∆). In this paper, it
is shown that for a ∆ being sequentially (S_{r}) and for every i
and j with 0 ≤ j ≤ i ≤ r−1, the equality h_{i,j}(∆) = h_{i,j}(Γ(∆)) holds true.
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