Discoveries / combinatorics

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 $m_0,\dots,m_{k−1}$, where $m_r$ 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 $k−1$.