Lexsegment ideals of Hilbert depth 1

Let $I\subset S=\text{KK}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ 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.