A variant of the Stanley depth for multisets

摘要

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, . . . , 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 n(k) with the bottom element deleted is (n-1) [k/2], which generalizes a result of Biro, 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]]. (C) 2019 Elsevier B.V. All rights reserved.