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

Mohammad Reza Pournaki Department of Mathematical Sciences, Sharif University of Technology Seyed Amin Seyed Fakhari Department of Mathematical Sciences, Sharif University of Technology Siamak Yassemi School of Mathematics, Statistics and Computer Science College of Science, University of Tehran

Algebraic Geometry Algebraic Topology and General Topology mathscidoc:1701.01018

Arkiv for Matematik, 51, (1), 185-196, 2011.3
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.
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@inproceedings{mohammad2011on,
  title={On the$h$-triangles of sequentially ($S$_{$r$}) simplicial complexes via algebraic shifting},
  author={Mohammad Reza Pournaki, Seyed Amin Seyed Fakhari, and Siamak Yassemi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634739633037},
  booktitle={Arkiv for Matematik},
  volume={51},
  number={1},
  pages={185-196},
  year={2011},
}
Mohammad Reza Pournaki, Seyed Amin Seyed Fakhari, and Siamak Yassemi. On the$h$-triangles of sequentially ($S$_{$r$}) simplicial complexes via algebraic shifting. 2011. Vol. 51. In Arkiv for Matematik. pp.185-196. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634739633037.
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