A variant of the Stanley depth for multisets

We define and study a variant of the Stanley depth which we call total depth for partially ordered sets (posets). This total depth is the most natural variant of Stanley depth from $?S_k?$ – 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 $?S_k?$ but not for such posets in general. We also prove that the total depth of the product of chains ${\mathit{n}}^k$ with the bottom element deleted is $(n?1)?k∕2?$, which generalizes a result of Biró, 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 $?S_k?$.