On scores, losing scores and total scores of hypertournaments

A k-hypertournament is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score si (losing score ri) of a vertex vi is the number of arcs containing vi in which vi is not the last element (in which vi is the last element). The total score of vi is defined as ti = si − ri. In this paper we obtain stronger inequalities for the quantities ∑ i∈I ri, ∑ i∈I si and ∑ i∈I ti, where I ⊆ {1, 2, . . . , n}. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong khypertournaments.