Tournament score sequences, Erdős–Ginzburg–Ziv numbers, and the Lévy–Khintchine method

We give a short proof of a recent result of Claesson, Dukes, Franklín and Stefánsson, connecting the number Sn of score sequences of length n and the Erdős–Ginzburg–Ziv numbers Nn from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erdős and Moser, and remarks by Kleitman added to an article of Moser regarding cyclical shifts of such paths. The connection between Sn and Nn is an instance of the Lévy–Khintchine formula from probability theory. We highlight the utility of such formulas by giving a short proof of Moser’s conjecture that Sn∼c4n/n5∕2, where c is described in terms of Nn.