Poset Entropy versus Number of Linear Extensions: the Width-$2$ Case

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.