  Let $I\subset S=\KK[x_1,...,x_n]$ be a lexsegment ideal, generated by
monomials of degree $d$. The main aim of this paper is to characterize when the
Hilbert depth of $I$ will be 1, in the standard graded case. In addition to
this, we will give an estimate of depth of squarefree monomial ideals,
generalizing a result of Popescu. We will also show that Stanley conjecture
holds for squarefree stable ideals, in the multigraded case.
