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      A New Depth Related to the Stanley Depth of Some Power Sets of Multisets

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          Abstract

          We define and study a new depth, related to the Stanley depth, for the partially ordered set (poset) of nonempty submultisets of a multiset. We find the new depth explicitly for any multiset with at most five distinct elements and provide an upper bound for the general case. On the other hand, the elements of a product of chains corresponds to the submultisets of a multiset. We prove that the new depth of the product of chains \(\bm{n}^k\backslash \bm{0}\) is \((n-1)\lceil{k\over 2}\rceil\). We also show that the new depth for any case of a multiset with \(n\) distinct elements can be determined if we know all interval partitions of the poset of nonempty subsets of \{1,2,...,\(n\)\}.

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          Linear diophantine equations and local cohomology

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            Author and article information

            Journal
            25 August 2009
            2010-11-11
            Article
            0908.3699

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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            13 pages
            math.CO

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