Score-Determined Induced Tournament Statistics: an All-Orders Classification

A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.

Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size?

Answer: Exactly the linear combinations of degree-multiplicity counts m₀,…,m_{k−1}, where mᵣ counts how many of the k chosen vertices have exactly r internal wins. These k functions satisfy one linear relation, so score-determined statistics have dimension k−1.