The theory of involutive divisions and an application to Hilbert function computations
Journal of Symbolic Computation
Generic and cogeneric monomial ideals
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals
Journal of Algebraic Combinatorics: An International Journal
Note: Interval partitions and Stanley depth
Journal of Combinatorial Theory Series A
On the Stanley depth of squarefree Veronese ideals
Journal of Algebraic Combinatorics: An International Journal
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In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed in a direct sum M = ⊕i = 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M.We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will derive a criterion which allows us to extract an upper bound on depth M from particular involutive bases. As a corollary we obtain that any monomial ideal M which possesses an involutive basis of this type satisfies Stanley's Conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of M. Moreover, we will show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithmic part for the computation of Stanley decompositions in these situations.