Let be two square free monomial ideals of a polynomial algebra over a field generated in degree , resp. . Almost always when contains precisely one variable, the other generators having degrees , if the Stanley depth of is then the usual depth of is too, that is the Stanley Conjecture holds in these cases.

Key Words: Monomial Ideals, Depth, Stanley depth.

2000 Mathematics Subject Classification: Primary: 13C15;

Secondary: 13F20, 13F55, 13P10.

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