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Perfect Sets and -Ideals
A square-free monomial ideal is called an {\it -ideal}, if both
and have the same
-vector, where (,
respectively) is the facet (Stanley-Reisner, respectively) complex related to
. In this paper, we introduce and study perfect subsets of and use
them to characterize the -ideals of degree . We give a decomposition of
by taking advantage of a correspondence between graphs and sets of
square-free monomials of degree , and then give a formula for counting the
number of -ideals of degree , where is the set of -ideals of
degree 2 in . We also consider the relation between an
-ideal and an unmixed monomial ideal.Comment: 15 page
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