Counting tournament score sequences

TL;DR

This paper proves Hanna's 2013 conjecture on a recursion for counting tournament score sequences, providing a closed formula and an efficient algorithm, thereby advancing understanding of tournament degree sequences.

Contribution

It confirms Hanna's conjecture by linking it to a main theorem that factorizes the generating function for score sequences, and introduces a closed formula and quadratic time counting algorithm.

Findings

Proves Hanna's conjecture on score sequence recursion

Derives a closed-form formula for counting score sequences

Develops a quadratic time algorithm for enumeration

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 $n$ 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.