The Art of Counting David M. Bressoud Macalester College St. Paul, MN BAMA, April 12, 2006 This Power Point presentation can be downloaded from

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

Building the next value from the previous values

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

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

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

+

“Pascal’s” triangle Published 1654

“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.

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

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

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

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

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

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

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

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

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

Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines Cut space by planes

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

Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines Cut space by planes

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

Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 Cut a plane by lines Cut space by planes ??

line by points plane by lines space by planes

line by points plane by lines space by planes

line by points plane by lines space by planes

line by points plane by lines space by planes

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

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?

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

Bill Mills Dave Robbins Howard Rumsey Institute for Defense Analysis

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

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

Monotone Triangle

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

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

n n A n = 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

n n A n = 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

1 1 2/ /3 3 3/ / / / / / /2 429

1 1 2/ /3 3 3/ /4 14 5/5 14 4/ / / / / / / / / /2 429

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

Numerators:

Conjecture 1: Numerators:

Conjecture 1: Conjecture 2 (corollary of Conjecture 1): For derivation, go to

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

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

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

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

+ q + 3q 2 + 6q q 4 + …

Generating function:

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

Totally Symmetric Self-Complementary Plane Partitions

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

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

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

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 ).

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

The End (which is really just the beginning) This Power Point presentation can be downloaded from