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Outcome of all the matches is unique

Source: 2018 RMM Shortlist C3

February 21, 2019
combinatoricsRMM Shortlistcounting

Problem Statement

NN teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order A=(a1≥a2≥⋯≥aN)\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N ) is known, as well as which team obtained which score. Find the number of sequences A\mathcal{A} such that the outcome of all matches is uniquely determined by this information.
[I]Proposed by Dominic Yeo, United Kingdom.
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