A Variant of the Stanley Depth for Multisets

TL;DR

This paper introduces a new variant called total depth for posets, compares it with Stanley depth, and derives formulas and bounds for multisets and product of chains, expanding understanding of poset depth measures.

Contribution

It defines total depth as a natural variant of Stanley depth for finite posets and explores its properties, including exact values and bounds for multisets and product chains.

Findings

Total depth equals Stanley depth for the poset of nonempty subsets of a set.

Total depth of product of chains is (n-1) times the ceiling of k/2.

Bounds and exact values for total depth are provided for multisets with up to five distinct elements.

Abstract

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 $\llbracket S_k\rrbracket$.