Counting tournament score sequences
Combinatorics
2023-01-18 v2
Abstract
The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with vertices is more than 100 years old (MacMahon 1920). In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.
Cite
@article{arxiv.2209.03925,
title = {Counting tournament score sequences},
author = {Anders Claesson and Mark Dukes and Atli Fannar Franklín and Sigurður Örn Stefánsson},
journal= {arXiv preprint arXiv:2209.03925},
year = {2023}
}