We say that a rank-unimodal poset P has rapidly decreasing rank numbers, or the RDR property, if above (resp. below) the largest ranks of P, the size of each level is at most half of the previous (resp. next) one. We show that a finite rank-unimodal, rank-symmetric, normalized matching, RDR poset of width w has a partition into w chains such that the sizes of the chains are one of two consecutive integers. In particular, there exists a partition of the linear lattices Ln (q) (subspaces of an n-dimensional vector space over a finite field, ordered by inclusion) into chains such that the number of chains is the width of Ln (q) and the sizes of the chains are one of two consecutive integers.
- Chain decompositions
- Generalized Füredi conjecture
- LYM property
- Linear lattices
- Normalized matching property