Some Sufficient Conditions for Finding a Nesting of the Normalized Matching Posets of Rank 3

TL;DR

This paper establishes sufficient conditions on rank numbers of rank 3 posets to verify Griggs's conjecture that normalized matching rank-unimodal posets are nested, extending known results beyond special cases.

Contribution

It provides new sufficient conditions for rank 3 posets to satisfy Griggs's conjecture, advancing understanding of nesting in graded posets.

Findings

Conditions for rank 3 posets to be nested

Extension of known cases of Griggs's conjecture

Progress towards the general case of the conjecture

Abstract

Given a graded poset $P$, consider a chain decomposition $\mathcal{C}$ of $P$. If $|C_1|\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\in \mathcal{C}$, then we say $\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$, and $P$ is said to be nested. In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested. This conjecture is proved to be true only for all posets of rank 2 [W:05], some posets of rank 3 [HLS:09,ENSST:11], and the very special cases for higher ranks. For general cases, it is still widely open. In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfies the Griggs's conjecuture.