Posets with seven linear extensions sortable by three comparisons

If every relation in a partial order also holds in a total order, then the total order is called linear extension of the partial order. The number of linear extensions is closely related with the number of minimum comparisons to sort the poset (D.E. Knuth, Sorting and Searching, 2nd ed., The Art of Computer Programming, Addison–Wesley, Reading, MA, 1998) [5]. We show that three comparisons suffice to sort any poset having seven linear extensions, and this result may speed up exhaustive computation to derive the lower bound of the minimum number of comparisons in sorting.

► Partially sorted configurations can be represented by posets.
► Three comparisons are shown enough to sort posets with seven linear extensions.
► Our result has been open for seven years.
► The result can be applied to compute the minimum number of comparisons in sorting.