1. The Art of Counting David M. Bressoud Macalester College St. Paul, MN BAMA, April 12, 2006 This Power Point presentation can be downloaded from www.macalester.edu/~bressoud/talks

2. 1.Review of binomial coefficients & Pascal’s triangle 2.Slicing cheese 3.A problem inspired by Charles Dodgson (aka Lewis Carroll)

3. Building the next value from the previous values

4. Choose 2 of them Given 5 objects How many ways can this be done?

5. Choose 2 of them Given 5 objects How many ways can this be done? ABCDE AB, AC, AD, AE BC, BD, BE, CD, CE, DE

6. Choose 2 of them Given 5 objects How many ways can this be done? ABCDE AB, AC, AD, AE BC, BD, BE, CD, CE, DE

7. +

8. + +++ +++ ++ +

9. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 “Pascal’s” triangle Published 1654

10. “Pascal’s” triangle from Siyuan yujian by Zhu Shihjie, 1303 CE Dates to Jia Xian circa 1100 CE, possibly earlier in Baghdad-Cairo or in India.

11. How many regions do we get if we cut space by 6 planes? George Pólya (1887–1985) Let Us Teach Guessing Math Assoc of America, 1965

14. How many regions do we get if we cut space by 6 planes? 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions

15. How many regions do we get if we cut space by 6 planes? 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions 4 planes: 15 regions

16. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7

17. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 124124

18. 1 2 3 4 5 6 7

19. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 12471247

20. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 12471247

22. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 1 2 4 7 11

24. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 1 2 4 7 11 16 22

25. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 1 2 4 7 11 16 22 Cut space by planes 1 2 4 8

27. 4th plane cuts each of the previous 3 planes on a line

28. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 1 2 4 7 11 16 22 Cut space by planes 1 2 4 8 15

29. 5th plane cuts each of the previous 4 planes on a line

30. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines 1 2 4 7 11 16 22 Cut space by planes 1 2 4 8 15 26 ??

31. 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1 line by points plane by lines space by planes

32. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

33. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

34. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

35. Number of regions created when space is cut by k planes:

36. Can we make sense of this formula? What formula gives us the number of finite regions? What happens in higher dimensional space and what does that mean?

37. Charles L. Dodgson aka Lewis Carroll “Condensation of Determinants,” Proceedings of the Royal Society, London 1866

38. Bill Mills Dave Robbins Howard Rumsey Institute for Defense Analysis

39. Alternating Sign Matrix: Every row sums to 1 Every column sums to 1 Non-zero entries alternate in sign

40. A 5 = 429 Alternating Sign Matrix: Every row sums to 1 Every column sums to 1 Non-zero entries alternate in sign

42. Monotone Triangle

44. 12345 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5

45. 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3

46. 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 322432542

47. 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3 14 22432542

48. 12345 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3 14723142623714 7 22432542

49. 12345 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3 147 105 23142623714 7 22432542

50. 12345 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3 147 4210513510542 23142623714 7 22432542

51. 12345 1234 1235 1245 1345 2345 123 124 125 134 135 145 234 235 345 12 13 14 15 23 24 25 34 35 45 1 2 3 4 5 3 147 4210513510542 429 23142623714 7 22432542

52. A 5 = 429 A 10 = 129, 534, 272, 700

53. A 5 = 429 A 10 = 129, 534, 272, 700 A 20 = 1436038934715538200913155682637051204376827212 = 1.43…  10 45

54. n123456789n123456789 A n 1 2 7 42 429 7436 218348 10850216 911835460 = 2  3  7 = 3  11  13 = 2 2  11  13 2 = 2 2  13 2  17  19 = 2 3  13  17 2  19 2 = 2 2  5  17 2  19 3  23

55. n123456789n123456789 A n 1 2 7 42 429 7436 218348 10850216 911835460 = 2  3  7 = 3  11  13 = 2 2  11  13 2 = 2 2  13 2  17  19 = 2 3  13  17 2  19 2 = 2 2  5  17 2  19 3  23

56. 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429 1 2 7 42 429 7436

57. 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429 +++

58. 1 2 3 2 7 14 14 7 42 105 135 105 42 429 1287 2002 2002 1287 429 +++ 1 1 3 1 2 5 1 2 4 5 1 2 3 4 5

59. 1 1 2/2 1 2 2/3 3 3/2 2 7 2/4 14 14 4/2 7 42 2/5 105 135 105 5/2 42 429 2/6 1287 2002 2002 1287 6/2 429

60. 1 1 2/2 1 2 2/3 3 3/2 2 7 2/4 14 5/5 14 4/2 7 42 2/5 105 7/9 135 9/7 105 5/2 42 429 2/6 1287 9/14 2002 16/16 2002 14/9 1287 6/2 429

61. 2/2 2/3 3/2 2/4 5/5 4/2 2/5 7/9 9/7 5/2 2/6 9/14 16/16 14/9 6/2

62. 1+1 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4 1+1 4+5 6+10 4+10 1+5 Numerators:

63. 1+1 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4 1+1 4+5 6+10 4+10 1+5 Conjecture 1: Numerators:

64. Conjecture 1: Conjecture 2 (corollary of Conjecture 1): For derivation, go to www.macalester.edu/~bressoud/talks

65. Richard Stanley, M.I.T. George Andrews, Penn State

66. length width n ≥ L 1 > W 1 ≥ L 2 > W 2 ≥ L 3 > W 3 ≥ … 1979, Andrews’ Theorem: the number of descending plane partitions of size n is

67. How many ways can we stack 75 boxes into a corner? Percy A. MacMahon

68. How many ways can we stack 75 boxes into a corner? Percy A. MacMahon # of pp’s of 75 = pp(75) = 37,745,732,428,153

69. + q + 3q 2 + 6q 3 + 13q 4 + …

70. Generating function:

71. For derivation, go to www.macalester.edu/ ~bressoud/talks A little algebra turns this generating function into a recursive formula:

72. Totally Symmetric Self-Complementary Plane Partitions

74. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is

75. 1989: William Doran shows equivalent to counting lattice paths 1990: John Stembridge represents the counting function as a Pfaffian (built on insights of Gordon and Okada) 1992: George Andrews evaluates the Pfaffian, proves Robbins’ Conjecture

76. 1996 Zeilberger publishes “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics Doron Zeilberger, Rutgers

77. 1996 Kuperberg announces a simple proof “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices Greg Kuperberg UC Davis Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

78. 1996 Zeilberger uses this determinant to prove the original conjecture “Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

79. The End (which is really just the beginning) This Power Point presentation can be downloaded from www.macalester.edu/~bressoud/talks