Landau's Theorem Revisited Again

We give a new proof of the sufficiency of Landau’s conditions for a non-decreasing sequence of integers to be the score sequence of a tournament. The proof involves jumping down a total order on sequences satisfying Landau’s conditions and provides a $O(n^2)$ algorithm that can be used to construct a tournament whose score sequence is any in the total order. We also compare this algorithm with to other algorithms that jump along this total order, one jumping down and one jumping up.