The poset of proper divisibility

Davide Bolognini; Antonio Macchia; Emanuele Ventura; Volkmar Welker

doi:10.1137/15M1049142

The poset of proper divisibility

Davide Bolognini, Antonio Macchia, Emanuele Ventura, Volkmar Welker

Department of Mathematics and Systems Analysis

University of Marburg

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Abstract

We study the partially ordered set P(a₁, ... , a_n) of all multidegrees (b₁, ... , b_n) of monomials x_b1 ₁ ... x^bn _n, which properly divide x_a1 1 ... x^an _n . We prove that the order complex Δ(P(a₁, ... , a_n)) of P(a₁, ... an) is (nonpure) shellable by showing that the order dual of P(a₁, ... , a_n) 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 language	English
Pages (from-to)	2093-2109
Number of pages	17
Journal	SIAM Journal on Discrete Mathematics
Volume	31
Issue number	3
DOIs	https://doi.org/10.1137/15M1049142
Publication status	Published - 2017
MoE publication type	A1 Journal article-refereed

Keywords

CL-shellability
Euler characteristic
Posets
Proper division
Simplicial homology

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AU - Bolognini, Davide

AU - Macchia, Antonio

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AU - Welker, Volkmar

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N2 - 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)).

AB - 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)).

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The poset of proper divisibility

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