Some Sufficient Conditions for Finding a Nesting of the Normalized Matching Posets of Rank 3
نویسندگان
چکیده
Given a graded poset P , consider a chain decomposition C of P . If |C1| ≤ |C2| implies that the set of the ranks of elements in C1 is a subset of the ranks of elements in C2 for any chains C1, C2 ∈ C, then we say 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 [11], some posets of rank 3 [10, 5], 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.
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