On the maximum number of Hamiltonian paths in tournaments (Q2725034)

For a tournament \(T\) with labelled vertices, \(P(T)\) (respectively \(C(T)\)) denote the numbers of Hamiltonian paths (respectively Hamiltonian cycles); the maximum over all tournaments \(T\) having \(n\) vertices are denoted by \(T(n)\) (respectively \(C(n)\)). A lower bound of \textit{T. Szele} [Kombinatorische Untersuchungen über den gerichteten vollständigen Graphen. (Hungarian. German summary). Mat. Fiz. Lapok 50, 223-256 (1943; Zbl 0061.41309)], that \(P(n)\geq\frac{n!}{2^{n-1}}\), is sharpened in Theorem 1: \(P(n)\geq(e-o(1))\frac{n!}{2^{n-1}}\) as \(n\to\infty\). A lower bound for \(C(n)\) is also investigated. It is observed that \textit{N. Alon} [Combinatorica 10, No. 4, 319-324 (1990; Zbl 0724.05036)] has shown that \(P(n)\leq O(n^{\frac 32}\cdot\frac{n!}{2^n})\), and that this bound has recently been improved by E. Friedgut and J. Kahn. ``It would be interesting to decide if \(P(n)=\Theta\left(\frac{n!}{2^n}\right)\)''.