6: 09 – Meeting ends, Lecture begins




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Pennsylvania State University

Math Club Meeting

Monday, January 23, 2006

6:00pm--- 106 McAllister

 

6:00 - Meeting begins

Recap from last meeting:

Mike had a brainstorm to step up the math lectures

Sellers will be presenting arithmetic properties of hyper m-ary partitions

Elections for a new secretary (Spring semester) - Sandi and Jenn


Bowling

Lauren - $10 for unlimited pizza and bowling

Snowtubing

Monday - 2 for 1

Tuesday - $10.50/person

Thursday $9.80/person

Friday (7-10) $10/person

 

James Sellers - has won 2 awards for faculty of the year, one at cedarville university (scholar) and distinguished undergraduate teaching from penn state math

54 journal articles, 90+ presentations, referee for 16 different journals

 

6:09 – Meeting ends, Lecture begins

Research done two years ago with Kevin Cortright(?) - published two papers together, one as he graduated, one after he left

Husband of one wife, and the father of five kids

Director of undergraduate mathematics at PSU


Partitions -

We'll talk about a certain family of partitions - hyperbinary

Definition with an example: n = 5

All the ways positive integers can be summed to get to 5 – 7 ways

5

4+1

3+2

3+1+1

2+2+1

2+1+1+1

1+1+1+1+1

Thus,

p(5) = 7

 

p(n) can be calculated easily

n p(n)

1 1

2 2

3 3

4 5

5 7

6 11

7 15

8 22

9 30

10 42

And so on…

 

First studied by Leonhard Euler 1707-1783

Leonhard Euler - arguably greatest mathematician of the century

Wrote about 500 mathematical papers

Made contributions to number theory, geometry, calculus, and graph theory

Also had 13 kids - only 5 of them survived adolescence

 

Contributing to the study of partitions - the study of pentagonal theorem

p(n) has a cool reoccurrence:

p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + p(n-12) + p(n-15)

1, 2, 5, 7, 12, and 15 are all numbers of the form (k(3k+/-1))/2) where k is a positive integer

It goes -, -, +, +, and so on

This does stop sooner or later

(k(3k+/-1))/2) are the pentagonal numbers

 

Euler: The Master of Us All by William Dunham

20-30 page history on Euler, and then 8-9 topical chapters

 

Srinivasa Ramanujan - 1887-1920 - phenomenal mathematician

Came up with a formula for p(n) and recognized some amazing results with p(n)

Realized that every fourth value in a group of (5 p(4), p(9), etc.) is a multiple of 5

5|p(5n+4) - mathematical formula stating the above

p(7n + 5) is p(5), p(12), etc… all multiples of 7

7|p(7n+5)

Could there be a pattern?

11|p(11n+6)

 

This began a study of what is now known as modular forms

This is the impetus of creating partitions for other numbers

 

And now for the main topic…

 

The function of the night!

Let h2(n) = number of hyperbinary partitions of n

= numbers of partitions of n into powers of 2 wherein each part can appear at most twice

 

Partitions of 5 that are hyperbinary…

5 X

4+1

3+1+1 X

2+2+1

2+1+1+1 X

1+1+1+1+1 X

So… h2(5) = 2

 

Bruce Resnick - studied various partition functions where he only wanted to use powers of 2

In the MAA monthly in 2000, Recounting the Rationals by Wilf and Caulkin

Proved a result about the positive rational numbers, that they are a countable set

When they did their work, they never thought of it as partitions - Kevin and James picked it up and studied it as its generating function:

n h2(n)

1 1

2 2

3 1

4 3

5 2

6 3

7 1

8 4

9 3

10 5

And so on…

 

h2(2n+1) = h2(2n) - h2(2n-1)

There seems to be symmetry around 2's

Number of values around 2 seems closely related to powers of 2

h2(2n+1) = h2(n)

h2(2^n - 1) = 1

h2(2n) = h2(n) + h2(n-1)

 

Are there any cool results for an arithmetic progression?

h2((2^(n-1))(2^(m)-1) = m(n-1)+1

Theorem: hs(3n+2) is always even.

2|h2(3n+2)

Corollary: the ONLY even values of h2 occur at 3n+2

Proof: by contradiction

Assume, to contradict (*), that N is the smallest positive integer such that h2(3N+2) is odd.

Case 1: Let N=2J+1

Then h2(3N+2)

=h2(3(2J+1)+2)

=h2(6J+5) - this is odd

=h2((6J+4)/2)

=h2(3J+2)

This is our contradiction, because J is smaller than N

Case 2: N=4J

Then h2(2N+2)

=h2(3(4J)+2)

=h2(12J+2)

=h2(6J+1)+h2(6J) - first is odd, second is even

=h2(3J)+h2(3J)+h2(3J-1)

=2h2(3J)+h2(3J-1) - the second must be odd, this second part is…:

h2(3(J-1+2), and J-1 is smaller than N

Case 3: Let N = 4J+2

Then h2(3N+2)

=h2(12J+8)

=h2(6J+4) + h2(6J+3)

=h2(3J+2)+h2(3J+1)+h2(3J+1)

=h2(3J+2)+2h2(3J+1) - h2(3J+2) has to be odd:

h2(3J+2), and J is smaller than N

 

Kevin and James defined a new function

hm(n) = numbers of partitions of n into powers of m used at most m times.

Theorem: For m≥3, hm(m^(j)n + m^(j-1)k) = j*hm(n)

Pick an m≥3, pick a j≥1, and pick a m-1≥k≥2

 

7:19 – Lecture ends

 

 

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