Sign-Balanced Posets

Let P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P. Call P sign-balanced if the number of linear extensions of P of positive sign is the same as the number of linear extensions of P of negative sign. In this paper we determine when the posets in a particular class are sign-balanced. When posets in this class are not sign-balanced, we determine the difference between the number of positive linear extensions and the number of negative linear extensions. One special case of this class is the product of an m-chain with an n-chain, m and n both >1. In this case, we show P is sign-balanced if and only if m≡n mod 2.