It was shown by W. V. Gehrlein\ and P. C. Fishburn\ [Comput. Math. Appl. {\bf 20} (1990), no.~2, 41--44; MR1062303 (91g:06004a)] that there are exactly 5 partially ordered sets on 9 elements where the linear extension majority (LEM) relation contains a cycle, and no such partially ordered sets on less than 9 elements. In this paper, an exact enumeration of all partially ordered sets whose LEM relation contains a cycle on at most 13 elements is given, including the lengths of the cycles. The smallest height-1 partially ordered sets with this property have 11 elements. All worst balanced partially ordered sets on at most 13 elements are listed. The counting algorithm makes use of the representation of a partially ordered set by its lattice of ideals.