Poset Entropy Versus Number of Linear Extensions: The Width-2 Case

Kahn and Kim (J. Comput. Sci. 51, 3, 390–399, 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 et al. (Combinatorica 33, 655–697, 2013). Here, we refine this last result in case P has width 2: we show that the constant can be replaced by 2?ε 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.