  Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$,
the entropy of the incomparability graph of $P$ (normalized by multiplying by
the order of $P$) and the base-$2$ logarithm of the number of linear extensions
of $P$ are within constant factors from each other. The tight constant for the
upper bound was recently shown to be $2$ by Cardinal, Fiorini, Joret, Jungers
and Munro (STOC 2010, Combinatorica). Here, we refine this last result in case
$P$ has width $2$: we show that the constant can be replaced by $2-\varepsilon$
if one also takes into account the number of connected components of size $2$
in the incomparability graph of $P$. Our result leads to a better upper bound
for the number of comparisons in algorithms for the problem of sorting under
partial information.
