A Note on Planar and Dismantlable Lattices

All lattices are assumed to be finite. Björner [2] has shown that a dismantlable (see Rival, [5]) lattice L is Cohen-Macaulay (see [6] for definition) if and only if L is ranked and interval-connected. A lattice is planar if its Hasse diagram can be drawn in the plane with no edges crossing. Baker, Fishburn and Roberts have shown that planar lattices are dismantlable, see [1]. Lexicographically shellable lattices are Cohen-Macaulay, see [3]. In a recent paper, [4], the author proved that a planar lattice L is lexicographically shellable if and only if L is rank-connected. s s s s s s s s s s s s s s s s ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ e e e e e e e e d d r r r r ¨ ¨ ¨ ¨ ¡ ¡ e e    ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ s s s s s s s s s s s s d d d d d d d d d d d d s s s s s s s s s s s d s d e e e e ¡ ¡ ¡ ¡ e e e e ¡ ¡ ¡ ¡ d d s s s s s s s s s d d d d d d e e ¡ ¡ (a) (b) (c) (d) Figure 1 We prove a conjecture of Björner that a dismantlable, rank-connected lattice is lexicographically shellable. We also show that an ranked and interval-connected lattice must be rank-connected. Hence, if L is a dismantlable lattice, L ranked and is interval-connected if and only if L is rank-connected. However, a rank-connected lattice need not be interval-connected. Figure 1(a) is a rank-connected lattice that is neither interval-connected nor planar. Not every dismantlable lattice is planar, see, for instance, Figure 1(d). In [4] it was conjectured that planar, rank-connected lattices are admissible (see Stanley, [7]). However, Figure 1(b) is a counterexample to that conjecture. Figure 1(c) is a planar, rank-connected lattice that is neither upper nor lower semi-modular. A lattice must have a least elementˆ0, and a greatest elementˆ1. A lattice that contains only a least element and a greatest element is trivial. A lattice is ranked if every maximal chain fromˆ0 tô 1 has the same length. For element x, r(x) is defined to be the length of