Degree sequences of <Emphasis Type="Italic">k</Emphasis>-multi-hypertournaments

Let n and k (n ≥ k > 1) be two non-negative integers. A k-multi-hypertournament on n vertices is a pair (V, A), where V is a set of vertices with |V| = n, and A is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V, A contains at least one (at most k!) of the k! k-tuples whose entries belong to S. The necessary and sufficient conditions for a non-decreasing sequence of non-negative integers to be the out-degree sequence (in-degree sequence) of some k-multi-hypertournament are given.