A Variant of the Stanley Depth for Multisets

We define and study a variant of the \emph{Stanley depth} which we call \emph{total depth} for partially ordered sets (posets). This total depth is the most natural variant of Stanley depth from $\llbracket S_k\rrbracket$ -- the poset of nonempty subsets of $\{1,2,\dots,k\}$ ordered by inclusion -- to any finite poset. In particular, the total depth can be defined for the poset of nonempty submultisets of a multiset ordered by inclusion, which corresponds to a product of chains with the bottom element deleted. We show that the total depth agrees with Stanley depth for $\llbracket S_k\rrbracket$ but not for such posets in general. We also prove that the total depth of the product of chains $\bm{n}^k$ with the bottom element deleted is $(n-1)\lceil{k/2}\rceil$, which generalizes a result of Bir{\'{o}}, Howard, Keller, Trotter, and Young (2010). Further, we provide upper and lower bounds for a general multiset and find the total depth for any multiset with at most five distinct elements. In addition, we can determine the total depth for any multiset with $k$ distinct elements if we know all the interval partitions of $\llbracket S_k\rrbracket$.