# Tutte seminar - Bruce Richmond

Friday, October 2, 2009 — 3:30 PM to 4:30 PM EDT

## Asymptotics of Some Partition Functions

Speaker: Bruce Richmond University of Waterloo Mathematics & Computer Building (MC) 5158

### Abstract:

Some recent results on the asymptotic analysis of partitions of integers will be discussed. First however some history of such problems will be briefly surveyed. The number, $p(m, n)$, of partitions of the integer $n$ into exactly $m$ parts will be studied. Euler showed that $$\sum_{n \ge 1} \sum_{m \ge 1} p(m, n)a^{m}z^{i} = \prod_{i \ge 1}\frac{1}{1 - az^{i}}.$$ There are many identities known between partition functions and much asymptotic analysis regarding these functions. A couple of examples will be described. Recently the number, $P(m,n)$ of plane partitions or two-dimensional partitions of $n$ with diagonal elements summing to $n$ has been studied. In this case $$\sum_{n \ge 1}\sum_{n \ge 1} P(m, n)a^{m}z^{n} = \prod_{i \ge 1} \frac{1}{\left(1 - az^{i}\right)^{i}}.$$ Using the saddle-point method with D. Panario and B. Young the asymptotic behaviour of $P(m, n)$ has been worked out. $P(m, n)$ satisfies a normal distribution. The details are unpleasantly intricate so only the barest sketch of the method which is widely applicable will be provided.

Location
MC - Mathematics & Computer Building
5158
200 University Avenue West

Waterloo, ON N2L 3G1

### December 2021

S M T W T F S
28
29
30
1
2
4
5
7
8
9
10
11
12
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
1
1. 2022 (66)
1. June (1)
2. May (13)
3. April (12)
4. March (18)
5. February (10)
6. January (13)
2. 2021 (103)
1. December (3)
2. November (7)
3. October (6)
4. September (12)
5. August (6)
6. July (10)
7. June (12)
8. May (7)
9. April (9)
10. March (13)
11. February (8)
12. January (10)
3. 2020 (119)
4. 2019 (167)
5. 2018 (136)
6. 2017 (103)
7. 2016 (137)
8. 2015 (136)
9. 2014 (88)
10. 2013 (48)
11. 2012 (39)
12. 2011 (36)
13. 2010 (40)
14. 2009 (40)
15. 2008 (39)
16. 2007 (15)