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> If $h(T)$ denotes the number of (directed) [Hamiltonian paths](https://en.wikipedia.org/wiki/Hamiltonian_path) in the [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) $T,$ what is the range of $h(T)$ as $T$ ranges over all (finite) tournaments $T$?
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By a classical theorem of [Rédei](https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_R%C3%A9dei) (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.



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> Is there an odd number $p\gt7$ such that no tournament has exactly $p$ Hamiltonian paths?
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The number of Hamiltonian paths in a tournament
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.


