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# Tom combinatorial choreography-4tu-ami-2017-06-23-reduced

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### Tom combinatorial choreography-4tu-ami-2017-06-23-reduced

1. 1. Combinatorial Choreography 4TU-AMI InterTU Study Day Tom Verhoeff Friday 23 June 2017
2. 2. Experiment 1 • 3 volunteers: A, B, C • Line up • Allowed moves: two neighbors swap places • Goal: present all possible orders exactly once / department of math & cs PAGE 223-06-17
3. 3. The 6 permutations of A B C ABCACB BAC BCA CAB CBA / department of math & cs PAGE 323-06-17
4. 4. Suitable Neigbor-Swap Order ABCACB BAC BCA CAB CBA / department of math & cs PAGE 423-06-17
5. 5. Experiment 2 • 4 volunteers: A, B, C, D • Line up • Allowed moves: two neighbors swap places • Goal: present all possible orders exactly once / department of math & cs PAGE 523-06-17
6. 6. Neighbor-Swap Graph for A B C D ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA / department of math & cs PAGE 623-06-17
7. 7. Graph Structure: Permutohedron for 1 2 3 4 / department of math & cs PAGE 723-06-17
8. 8. Miter Joints / department of math & cs PAGE 823-06-17 Joint angle = 60.0° Edges nicely meet across the joint
9. 9. Miter Joints in Space Are Problematic / department of math & cs PAGE 923-06-17
10. 10. Skew Miter Joints / department of math & cs PAGE 1023-06-17
11. 11. Hamilton Cycle on Truncated Octahedron / department of math & cs PAGE 1123-06-17
12. 12. Presenting Permutations by Neighbor Swaps • Always possible • Already known in 17th century • Change ringing • Nice recursive algorithm • Hamilton cycle • First and last permutation differ by a neighbor swap / department of math & cs PAGE 1223-06-17
13. 13. Origin of Problem • IMO 2011, The Netherlands • IMO 2009, Germany • Opening Ceremony: 100+ teams / department of math & cs PAGE 1323-06-17
14. 14. Opening Ceremony IMO 2009, Germany / department of math & cs PAGE 1423-06-17
15. 15. Experiment 3: Bit Sequences • 4 volunteers 0, 0, 1, 1 • Line up • Allowed moves: two distinct neighbors swap places • Goal: present all possible orders exactly once / department of math & cs PAGE 1523-06-17
16. 16. Neighbor-Swap Graph for 0 0 1 1 0011 0101 0110 1001 1010 1100 / department of math & cs PAGE 1623-06-17
17. 17. Neighbor-Swap Graph for 0 0 0 1 1 1 000111 001011 001101 001110 010011 010101 010110 011001 011010 011100 100011 100101 100110 101001 101010 101100 110001 110010 110100 111000 / department of math & cs PAGE 1723-06-17
18. 18. Hamilton Path for 0 0 0 1 1 1 000111 001011 001101 001110 010011 010101 010110 011001 011010 011100 100011 100101 100110 101001 101010 101100 110001 110010 110100 111000 / department of math & cs PAGE 1823-06-17
19. 19. What Is Known for 0a 1b • 1984 (Eades, Hickey, Read): Hamilton path exists iff • a ≤ 1 or b ≤ 1 (trivial: graph is chain), or • both a and b are odd • “Only if”: later • “If”: Constructive proof through recursive algorithm • Alternatives by Cor Hurkens, Ivo van Heck, Tom V. / department of math & cs PAGE 1923-06-17
20. 20. More Than Two ‘Colors’: 0a 1b 2c … • 1992 (Stochawiak): Hamiltonian path exists iff • Trivial, or • At least two odd color counts • Hamilton cycle exists, unless • Graph is linear • Two colors, one of which has odd count • Three colors with counts (even, 1, 1) / department of math & cs PAGE 2023-06-17
21. 21. D. H. Lehmer’s Conjecture (1965) • A –possibly imperfect– Hamilton path exists (always) • Imperfect Hamilton path: allows (order 1) spurs / department of math & cs PAGE 2123-06-17 A B C
22. 22. “The Spurs of D. H. Lehmer” (Verhoeff, 2016) • Elegant proof that ≥ two odd color counts are needed • Elegant proof for binary case of Lehmer’s Conjecture • For general case: cycle cover exists / department of math & cs PAGE 2223-06-17
23. 23. Permutations • Inversion: out-of-order pair • 2 1 1 0 contains 5 inversions • Parity: number of inversions modulo 2 • 2 1 1 0 has odd parity • Neighbor swap changes inversion count by 1 • Flips the parity • Neighbor-Swap graph is bipartite: parity alternates / department of math & cs PAGE 2323-06-17
24. 24. 0011 0101 0110 1001 1010 1100 / department of math & cs PAGE 2423-06-17 000111 001011 001101 001110 010011 010101 010110 011001 011010 011100 100011 100101 100110 101001 101010 101100 110001 110010 110100 111000
25. 25. Number of Even Minus Odd Permutations • M(a, b, c, …) = number of permutations of 0a 1b 2c … • Multinomial coefficient: a + b + c + … over a, b, c, … • D(a, b, c, …) = number of even minus odd perm’s • D(a, b, c, …) = M(a÷2, b÷2, c÷2, …) if ≤ 1 odd count, otherwise 0 / department of math & cs PAGE 2523-06-17
26. 26. Stutter Permutations • Partition from left into pairs: • e1 e2 | e3 e4 | … | e2i-1 e2i | … • All e2i-1 = e2i • Possibly single item at end • E.g. 1 1 0 0 2 2 3 • Stutter permutations are even • Stutter permutation has ≤ 1 odd color counts • Non-stutter permutations can be paired even-to-odd • 1 1 0 0 2 3 2 ↔ 1 1 0 0 3 2 2 • D(a, b, c, …) = number of stutter permutations / department of math & cs PAGE 2623-06-17
27. 27. Tom’s Conjecture • The stutter permutations can serve as spur tips • There exists a Hamilton cycle on the non-stutter permutations, except … (see Stachowiak) • Proven for binary case • Cycle cover for general case / department of math & cs PAGE 2723-06-17
28. 28. Combining 2 paths with spurs into 1 path w/o / department of math & cs PAGE 2823-06-17
29. 29. / department of math & cs PAGE 2923-06-17
30. 30. / department of math & cs PAGE 3023-06-17
31. 31. / department of math & cs PAGE 3123-06-17
32. 32. Braids / department of math & cs PAGE 3223-06-17
33. 33. KNAW Mingler Netwerk for Science & Arts • Cooperation with choreographer Roos van Berkel • Looking for funding to create Lehmer’s Dance / department of math & cs PAGE 3323-06-17