Loopy, Hankel, and Combinatorially Skew-Hankel Tournaments

We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type.