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

, if m = n. It is a classical result of Landau [5], (see Moon [6], Ford and Fulkerson [3]) that the above conditions are both necessary and sufficient for existence of a complete digraph on [n] whose (size-ordered) out-degree sequence is s . Such a digraph is interpreted as an outcome of a round-robin tournament, with [n] as the set of players, and s comprised by scores put in order of increase. Let Sn denote the total number of all such s . In a seminal paper [4] Kleitman and Winston were able to show that for some absolute (positive) constants c1, c2 (1.2) c1 4 n5/2 ≤ Sn ≤ c2 4 n2 , thus improving the earlier bounds due to Erdös and Moser, see [6], by factors n and n respectively. Kleitman and Winston demonstrated that an upper bound in (1.2) could be upgraded to a best possible bound (1.3) Sn ≤ c3 4 n5/2 ,