Undergraduate Mathematics Students
Thursday, February 5: Binomial coefficients and beyond
Come enjoy this week's SUM Series talk:
Binomial coefficients and beyond
Patricia Hersh
Thursday, February 5, 2009
3:00--3:50 p.m.
Harrelson 330
If you multiply out the product (x+y)n, the coefficient of xkyn-k is known as a binomial coefficient.
For instance, (x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4.
The coefficient of $xkyn-k counts subsets of {1,...,n} of size k, so for instance there are 6 subsets of {1,2,3,4} of size 2.
This coefficient also counts the paths from (0,0) to (2,2) in the plane comprised of steps (1,0) and (0,1).
But what if x and y don't commute?
What if yx = qxy? It turns out the coefficients now count paths in a more refined way, with the coefficient of qAxkyn-k counting paths from (0,0) to (k,n-k) having area A below the path.
I will discuss this result and some other interesting properties of this q-analogue of binomial coefficients that can be proven using a mixture of linear algebra and group theory or in an especially slick way using the representation theory of sl2.
A sad announcement: SUM Series pizza has become a casualty of the economic downturn. The pizza money is still in the SUM budget, but the State of NC says we're not allowed to spend it. (This is part of a statewide freeze on certain types of expenditures.) We're forced to test whether people are coming for the math or for the pizza!
A happier announcement: Our speaker, Patricia Hersh, has graciously offered to bring chocolate this Thursday, so we won't starve!
Check out the SUM Series website for more information on the SUM Series.
TELL YOUR FRIENDS!
Posted at 11:43AM Feb 02, 2009 by nreadin in General | Comments[0]