Order’s Problem List

Abstract

For a finite ordered set (X,<), let N(S) stand for the number of linear extensions with property S. Thus, N=N(O) is the number of all linear extensions for S=O, that is no constraints at all. Let \( p(S) = \frac{{N(S)}}{N} \), a counting probability measure according to which all linear extensions are assumed to be equally likely. N(xy) is the number of linear extensions in which x precedes y. Then p(xy)+p(yx)=1 and p(xy)=1 if and only if x<y.