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$f$-Ideals
A square-free monomial ideal 
$I$ of 
$k[x_{1},\ldots ,x_{n}]$ is said to be an 
$f$-ideal if the facet complex and non-face complex associated with 
$I$ have the same 
$f$-vector. We show that 
$I$ is an 
$f$-ideal if and only if its Newton complementary dual 
$\widehat{I}$ is also an 
$f$-ideal. Because of this duality, previous results about some classes of 
$f$-ideals can be extended to a much larger class of 
$f$-ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for 
$f$-vectors of simplicial complexes.
