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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,...,$$n$$\}.

### Most cited references1

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### Linear diophantine equations and local cohomology

(1982)
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### Author and article information

###### Journal
25 August 2009
2010-11-11
###### Article
0908.3699