Session 23 of 28 · 27 May 2025 · 1 h 05 min

The Number of Score Vectors

Combinatorial analysis of possible score vectors

Notes for this session are on their way: they are being written from the transcript of the recording.

Topics on Tournaments

5 May Introduction Basic definitions and fundamental concepts of tournaments ▶ 1 h 08 min notes
6 May Irreducible Tournaments Exploring irreducible tournaments and their properties ▶ 58 min
7 May Strong Tournaments Understanding strong tournaments and their characteristics ▶ 33 min
8 May Cycles in a Tournament Analysis of cycles and their significance in tournaments ▶ 37 min
9 May Strong Subtournaments of a Tournament Examining strong subtournaments and their relationships not recorded
10 May The Distribution of 3-cycles in a Tournament Statistical analysis of 3-cycles in tournament structures not recorded
11 May Transitive Tournaments Exploring transitive tournaments and their unique properties ▶ 41 min
12 May Sets of Consistent Arcs in a Tournament Understanding consistent arc sets and their implications ▶ 1 h 00 min
13 May The Parity of the Number of Spanning Paths of a Tournament Examining parity properties of spanning paths ▶ 1 h 15 min
14 May The Maximum Number of Spanning Paths of a Tournament Analysis of maximum spanning paths in tournament structures not recorded
15 May An Extremal Problem Exploring extremal problems in tournament theory ▶ 1 h 09 min
16 May The Diameter of a Tournament Understanding tournament diameter and its implications ▶ 42 min
17 May The Powers of Tournament Matrices Examining matrix representations of tournaments and their powers not recorded
18 May Scheduling a Tournament Practical applications in tournament scheduling ▶ 1 h 25 min
19 May Ranking the Participants in a Tournament Methods and algorithms for ranking tournament participants ▶ 1 h 03 min
20 May The Minimum Number of Comparisons Necessary to Determine a Transitive Tournament Optimization problems in transitive tournaments not recorded
21 May Universal Tournaments Exploring the concept and properties of universal tournaments ▶ 1 h 02 min
22 May Expressing Oriented Graphs as the Union of Bilevel Graphs Representation theory for oriented graphs in tournaments ▶ 1 h 02 min
23 May Oriented Graphs Induced by Voting Patterns Applications of tournament theory to voting systems ▶ 46 min
24 May Oriented Graphs Induced by Team Comparisons Applications of tournament theory to sports and competitions ▶ 59 min
25 May Criteria for a Score Vector Understanding the mathematical criteria for valid score vectors not recorded
26 May Score Vectors of Generalizations of Tournaments Extending score vector concepts to generalized tournaments ▶ 56 min
27 May The Number of Score Vectors Combinatorial analysis of possible score vectors ▶ 1 h 05 min
28 May The Largest Score in a Tournament Extremal analysis of maximum scores in tournaments ▶ 1 h 32 min
29 May A Reversal Theorem Exploring reversal properties in tournament theory ▶ 1 h 01 min
30 May Tournaments with a Given Automorphism Group Algebraic analysis of tournament symmetries ▶ 52 min
31 May The Group of the Composition of Two Tournaments Understanding compositional properties of tournament groups ▶ 58 min
1 Jun The Maximum Order of the Group of a Tournament Exploring maximum order properties in tournament groups ▶ 42 min