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A New Connection Between the Triangles of Stirling and Pascal
Craig Bauer
York College of PA
Pascal’s Triangle
Triangular Numbers
Tetrahedral Numbers
Pentatop Numbers
Row Sums – Powers of 2
Fibonacci Numbers
Hockey Stick Patterns
Picture from http://ptri1.tripod.com/
Generating Function
(x+1)n
Shaded Modulo 2
Image from http://wyvern-community.school.hants.gov.uk/sierpinski.htm
Mod 2 with More Rows
Images from http://www.pittstate.edu/math/Cynthia/pascal.html
Mod 3
Mod 4
Mod 5
Mod 6
Mod 7
Investigate for Yourself!
http://binomial.csuhayward.edu/applets
/appletGasket.html
Perfect Numbers
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 2n –1(2n – 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!
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.
A Simple Pattern
For three points, we get 4 regions.
For four points, we get 8 regions.
For five points, we get 16 regions.
Make a Prediction!
We have the sequence 1, 2, 4, 8, 16, …
What will the next term be?
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.
*A000127
Partial Row Sums
Some Formulas
Recursive
Non-recursive
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!)
“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
Stirling’s Triangle
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
• Answer #2 – How can we express the nth power of x as a sum of “factorials”?
• Example: x4
x4 = 1x(x – 1)(x – 2)(x – 3) +6x(x – 1)(x – 2) +7x(x – 1) +1x
Coefficients are: 1 6 7 1
Where Does it Come From?
Row Sums – Bell Numbers
Exponential Generating Function
kx
n
n
ekn
xknS1
!
1
!
),(
0
Some Formulas
Recursive
Non-recursive
Stirling’s Triangle mod 2
Stirling’s Triangle mod 3
Eighty rows of Stirling Numbers of the second kind mod 3From http://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
Stirling’s Triangle mod 4
Stirling’s Triangle mod 5
And now for something completely different…
Upper TriangularPartial Permutation
Matrices
j
ih
gfe
dcba
000
00
0
At most a single 1 in any row or column.
No 1s below the main diagonal.
Examples
1 by 1
only 2 possibilities
0 1
Examples
2 by 2
only 5 possibilities
Examples
3 by 3
only 15 possibilities
Sorted by Dimension & Rank
A New Twist
k=1 k=2 k=3
Insist on k extra diagonals of 0s
above the main diagonal.
j
h
ge
dca
000
000
00
0
j
h
e
da
000
000
000
00
j
h
e
a
000
000
000
000
Counting by Rank (k=1)
A Simple Rule
P(n,k) = P(n – 1,k) +(n – k)P(n – 1,k – 1) +P(n – 2,k – 2)
Fibonacci Numbers
Number of n by n matrices of rank n-1 is
Triangular Numbers
n + tn–2
Number of n by n matrices of rank 1 is
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.
The Big Picture
Some Nice General Results
For n ≤ 2k + 2,
Not as General (or Nice)
For n = 2k + 3,
Not Elegant!For n = 2k + 4,
lim k →∞ ?
Pascal’s Triangle!
k=1 Triangle mod 2
k=1 Triangle mod 3
k=1 Triangle mod 4
k=1 Triangle mod 5
Counting by Rank (k=2)
Molinar’s Conjecture for k = 2
Sullivan’s Result
Pattern?
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.
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}.
