Counting Linear Extensions: Parameterizations by Treewidth

We consider the [Formula: see text] -complete problem of counting the number of linear extensions of a poset [Formula: see text] ; a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of [Formula: see text] parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that [Formula: see text] is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that [Formula: see text] becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.