Confirming the Kleitman–Winston Conjecture on the Largest Coefficient in a q-Catalan Number

In 1983 Kleitman and Winston conjectured that the largest coefficient in an nth q-Catalan number is of order O(4n/n3/2). Assuming its truth, they proved that the total number of n-tournament score sequences is O(4n/n5/2), thus matching their own lower bound. Our purpose is to confirm the conjecture.