1. A New Connection Between the Triangles of Stirling and Pascal Craig Bauer York College of PA

2. Pascal’s Triangle

3. Triangular Numbers

4. Tetrahedral Numbers

5. Pentatop Numbers

6. Row Sums – Powers of 2

7. Fibonacci Numbers

8. Hockey Stick Patterns Picture from http://ptri1.tripod.com/

9. Generating Function (x+1) n

10. Shaded Modulo 2 Image from http://wyvern-community.school.hants.gov.uk/sierpinski.htmhttp://wyvern-community.school.hants.gov.uk/sierpinski.htm

11. Mod 2 with More Rows Images from http://www.pittstate.edu/math/Cynthia/pascal.html

12. Mod 3

13. Mod 4

14. Mod 5

15. Mod 6

16. Mod 7

17. Investigate for Yourself! http://binomial.csuhayward.edu/applets /appletGasket.html

18. Perfect Numbers

19. Disclaimer But the sequence of the number of elements in each white triangle began with 1 and this isn’t a perfect number! That’s true, Pascal’s triangle doesn’t always yield perfect numbers in this manner, but every even perfect number does appear somewhere in this sequence. This is because the number of elements in each white triangle is given by 2 n –1 (2 n – 1). With n = 1, we get 1. Making n = 2 or 3 gives 6 and 28, respectively. Every even perfect number is of this form, but not every number of this form is perfect. What about odd perfect numbers? Are there any? Nobody knows!

20. A Simple Pattern For just one point, we cannot draw any lines, so have 1 region. For two points, we may draw a line to get 2 regions.

21. A Simple Pattern For three points, we get 4 regions. For four points, we get 8 regions. For five points, we get 16 regions.

22. Make a Prediction! We have the sequence 1, 2, 4, 8, 16, … What will the next term be?

23. WRONG! References *I first saw the problem described above in The (Fabulous) Fibonacci Numbers by Alfred S. Posamentier and Ingmar Lehmann, Prometheus Books, June 2007. *A000127A000127

24. Partial Row Sums

25. Some Formulas Recursive Non-recursive

27. George Lilley, Pascal’s Arithmetic Triangle, American Mathematical Monthly, Vol. 1, No. 12, Dec., 1894, p.426. (Well over 200 years after Pascal’s death!)

28. “This representation comes from China. It dates from a book of 1303 CE written by Chu Shï-kié. The earliest known use of the pattern was by Yang Hui, whose books date from 1261 & 1275 CE. Chu Shï-kié refers to the triangle as already being old. Jamshid Al-Kashi, who died around 1436 CE, was an astronomer at the court of Ulugh Beg in Samarkand in the 15th Century. Al-Kashi was the first known Arabic author to consider 'Pascal's' Triangle.” picture and text from: http://www.bbc.co.uk/education/asguru/maths/14statistics/03binomialdistribution/8binomialdistribution/index.shtml http://www.bbc.co.uk/education/asguru/maths/14statistics/03binomialdistribution/8binomialdistribution/index.shtml

29. Stirling’s Triangle

30. Where Does it Come From? Answer #1 – In how many ways can a set of n distinct objects be split into k nonempty disjoint subsets? Example: n=4 k=1 k=2 k=3 k=4

31. Answer #2 – How can we express the nth power of x as a sum of “factorials”? Example: x 4 x 4 = 1x(x – 1)(x – 2)(x – 3) +6x(x – 1)(x – 2) +7x(x – 1) +1x Coefficients are: 1671 Where Does it Come From?

32. Row Sums – Bell Numbers

35. Exponential Generating Function

36. Some Formulas Recursive Non-recursive

37. Stirling’s Triangle mod 2

38. Stirling’s Triangle mod 3

39. Eighty rows of Stirling Numbers of the second kind mod 3 From http://www.cecm.sfu.ca/~loki/Papers/Numbers/node7.htmlhttp://www.cecm.sfu.ca/~loki/Papers/Numbers/node7.html Note: This illustration starts with n heap 0 = 0 for each row. Stirling’s Triangle mod 3

40. Stirling’s Triangle mod 4

41. Stirling’s Triangle mod 5

42. And now for something completely different…

43. Upper Triangular Partial Permutation Matrices At most a single 1 in any row or column. No 1s below the main diagonal.

44. Examples 1 by 1 only 2 possibilities

45. Examples 2 by 2 only 5 possibilities

46. Examples 3 by 3 only 15 possibilities

47. Sorted by Dimension & Rank

48. A New Twist k=1 k=2 k=3 Insist on k extra diagonals of 0s above the main diagonal.

49. Counting by Rank (k=1)

50. A Simple Rule P(n,k) = P(n – 1,k) +(n – k)P(n – 1,k – 1) +P(n – 2,k – 2)

51. Fibonacci Numbers Number of n by n matrices of rank n-1 is

52. Triangular Numbers n + t n–2 Number of n by n matrices of rank 1 is

53. More Triangles We have an infinite sequence of triangles. They are all distinct. Comparing terms in a fixed location of the triangles always gives a decreasing (convergent) sequence.

54. The Big Picture

55. Some Nice General Results For n ≤ 2k + 2,

56. Not as General (or Nice) For n = 2k + 3,

57. Not Elegant! For n = 2k + 4,

59. lim k →∞ ? Pascal’s Triangle!

60. k=1 Triangle mod 2

61. k=1 Triangle mod 3

62. k=1 Triangle mod 4

63. k=1 Triangle mod 5

64. Counting by Rank (k=2)

65. Molinar’s Conjecture for k = 2

67. Sullivan’s Result

68. Pattern?

69. References 1. Bauer, C., Triangular Monoids and an Analog to the Derived Sequence of a Solvable Group, International Journal of Algebra and Computation, Vol. 10, No. 3 (2000) pp. 309-321. 2. Borwein, D., Rankin, S., and Renner, L., Enumeration of Injective Partial Transformations, Discrete Mathematics, Vol. 73, 1989, p. 291- 296.

70. From Wikipedia: Cereal Box Problem The Stirling numbers of the second kind can represent the total number of ways a person can collect all prizes after opening a given number of cereal boxes. For example, if there are 3 prizes, and one opens three boxes, there is S(3,3), or 1 way to win, {1,2,3}. If 4 boxes are opened, there are 6 ways to win S(4,3); {1,1,2,3}, {1,2,1,3}, {1,2,3,1}, {1,2,2,3}, {1,2,3,2}, {1,2,3,3}.