Abstract
We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex. We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators, we obtain a partial result.
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Notes
The assumption in [14] that \(\phi \) is a map of the lcm-semilattices is equivalent to the assumption that \(\phi \) is injective on \(\hat{0}_L\), because the semilattices in our situation are \(L \setminus \{\hat{0}_L\}\) and \(L' \setminus \{\hat{0}_{L'}\}\).
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Acknowledgments
The author would like to thank Winfried Bruns for suggesting the method described in 7.2.1 and Markus Spitzweck for discussing the proof of Theorem 4.3. Moreover, the author thanks Bogdan Ichim, Francesco Strazzanti, Richard Sieg and Mihai Cipu for pointing out many typos in an earlier version of this article. Further, I would like to thank the two anonymous reviewers for several helpful hints and suggestions.
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The author was partially supported by the German Research Council DFG-GRK 1916.
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Katthän, L. Stanley depth and simplicial spanning trees. J Algebr Comb 42, 507–536 (2015). https://doi.org/10.1007/s10801-015-0589-y
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DOI: https://doi.org/10.1007/s10801-015-0589-y