TITLE: The number of Hamiltonian paths in a tournament
QUESTION [11 upvotes]: If $h(T)$ denotes the number of (directed) Hamiltonian paths in the tournament $T,$ what is the range of $h(T)$ as $T$ ranges over all (finite) tournaments $T$?

By a classical theorem of Rédei (László Rédei, Ein kombinatorischer Satz, Acta Litteraria Szeged 7 (1934), 39–43), every tournament has an odd number of Hamiltonian paths. Moreover, one can verify that there is no tournament with exactly $7$ Hamiltonian paths.

Is there an odd number $p\gt7$ such that no tournament has exactly $p$ Hamiltonian paths?

REPLY [6 votes]: This sounds bizarre, but my computations suggest that 21 might be a possibility..
I checked the tournaments on up to 9 vertices (10 vertices is under way) and got the following results:

by 6 vertices, every odd number (except 7 and 21) less than 33 had occurred at least once
by 7 vertices, every odd number (except 7 and 21) less than 63 had occurred at least once
by 8 vertices, every odd number (except 7 and 21) less than 611 had occurred at least once
by 9 vertices, every odd number (except 7 and 21) less than 2883 had occurred at least once

So there is an obvious conjecture based on those numbers...
Edit (Additional data): By 10 vertices, every odd number (not 7, 21) less than 14685 has occurred. So now I am convinced that there is no tournament with exactly 21 hamilton paths
Edit (Even more additional data): By 11 vertices, every odd number (not 7, 21) less than 80557 has occurred as the number of hamiltonian paths of a tournament on at most 11 vertices.