We study the partially ordered set P(a1,...,an) of all multidegrees (b1,...,bn) of monomials x(1)(b1)...x(n)(bn), which properly divide x(1)(a1)...x(n)(an). We prove that the order complex Delta(P(a(1),...,a(n))) of P(a(1),...a(n)) is (non-pure) shellable, by showing that the order dual of P(a(1),...,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 Delta(P(a1,a2)). Furthermore, we give a succinct formula for the Euler characteristic of Delta(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. - ELETTRONICO. - 31:3(2017), pp. 2093-2109. [10.1137/15M1049142]
The poset of proper divisibility
Bolognini D.;
2017-01-01
Abstract
We study the partially ordered set P(a1,...,an) of all multidegrees (b1,...,bn) of monomials x(1)(b1)...x(n)(bn), which properly divide x(1)(a1)...x(n)(an). We prove that the order complex Delta(P(a(1),...,a(n))) of P(a(1),...a(n)) is (non-pure) shellable, by showing that the order dual of P(a(1),...,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 Delta(P(a1,a2)). Furthermore, we give a succinct formula for the Euler characteristic of Delta(P(a1,a2)).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
