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

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,...,$n$\}.