There is a webcam in my office that I use to take photos of the whiteboard. This setup was inspired by Dror Bar Natan’s Blackboard Shots. You can read more about the setup here. If you know of anyone else with a similar setup, please let me know.
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For positive integers $m$ and $n$, let $f(m, n)$ denote the number of $n$-tuples $(x_1, x_2, \dots , x_n)$ of integers such that $|x_1|+|x_2|+ \cdots +|x_n| \leq m$. Show that $f(m, n) = f(n, m)$.
The photo on the blackboard verifies that $f(2,3) = f(3,2)$.
![A photo of a whiteboard titled: Ideals and Principal Ideals: Are all ideals in Z[x,y] actually principal?](/images/office-camera/2023-05-30T12:01:54-04:00.jpeg "Ideals and Principal Ideals: Are all ideals in Z[x,y] actually principal?")
I’m working with Eric Vandendriessche and Alfredo Braunstein to understand a bit about how the string figure calculus acts of canonical linear sequences. This digram shows a simple example of the type of question we want to be able to answer.
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