We study the partially ordered set P(a1, ... , an) of all multidegrees (b1, ... , bn) of monomials xb11 ... xbnn, which properly divide xa1 1 ... xann . 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)).

The poset of proper divisibility / Bolognini, D.; Macchia, A.; Ventura, E.; Welker, V.. - In: SIAM JOURNAL ON DISCRETE MATHEMATICS. - ISSN 0895-4801. - 31:3(2017), pp. 2093-2109. [10.1137/15M1049142]

The poset of proper divisibility

Ventura E.;
2017

Abstract

We study the partially ordered set P(a1, ... , an) of all multidegrees (b1, ... , bn) of monomials xb11 ... xbnn, which properly divide xa1 1 ... xann . 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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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2958161