Write $ \mathbf{Sym}(X) $ for the set of permutations of $ X $ . A total order on a set $ X $ is an order $ \leq $ such that for all $ x, y \in X $ , either $ x \leq y $ or $ y \leq x $ ; so a total order on a finite set amounts to a way of placing its elements in sequence. Write $ \mathbf{Ord}(X) $ for the set of total orders on $ X $ .