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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
$\Delta$ be a $(d-1)$-dimensional simplicial complex with $\Gamma(\Delta)$
as its algebraic shifting. Also let $(h_{i,j}(\Delta))_{0\leq j\leq i\leq
d}$ be the $h$-triangle of $\Delta$ and $(h_{i,j}(\Gamma(\Delta)))_{0\leq
j\leq i\leq d}$ be the $h$-triangle of $\Gamma(\Delta)$. In this paper, it
is shown that for a $\Delta$ being sequentially $(S_r)$ and for every $i$
and $j$ with $0\leq j\leq i\leq r-1$, the equality $h_{i,j}(\Delta)= h_{i,
j}(\Gamma(\Delta))$ holds true.
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