On the consistency of score sheets of a round-robin football tournament

Abstract

In this paper we introduce the submonoids $\mathscr{R}_n$, resp. $\mathscr{C}_n$, of the monoid $\mathscr{M}_n$ of ordered score sheets of a robin-round tournament played by $n$ teams for which the order is preserved after the leader team is disqualified, resp. all principal submatrices preserve the given ordering. We study (using both theoretical and computational methods) the most important invariants of these monoids, namely the Hilbert basis, the multiplicity, the Hilbert series and the Hilbert function. In particular we give a general description of the Hilbert basis of $\mathscr{R}_n$ and we show that $\mathscr{C}_n$ is Gorenstein for $n>2$.