The poset of proper divisibility

Davide Bolognini, Antonio Macchia, Emanuele Ventura, Volkmar Welker

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Abstract

We study the partially ordered set P(a1, ... , an) of all multidegrees (b1, ... , bn) of monomials xb1 1 ... xbn n, which properly divide xa1 1 ... xan n . We prove that the order complex Δ(P(a1, ... , an)) of P(a1, ... an) is (nonpure) shellable by showing that the order dual of P(a1, ... , an) is CL-shellable. Along the way, we exhibit the poset P(4, 4) as a new example of a poset with CL-shellable order dual that is not CL-shellable itself. For n = 2, we provide the rank of all homology groups of the order complex δ(P(a1, a2)). Furthermore, we give a succinct formula for the Euler characteristic of δ(P(a1, a2)).

Original languageEnglish
Pages (from-to)2093-2109
Number of pages17
JournalSIAM Journal on Discrete Mathematics
Volume31
Issue number3
DOIs
Publication statusPublished - 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • CL-shellability
  • Euler characteristic
  • Posets
  • Proper division
  • Simplicial homology

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