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# [theory-seminar] Theory Lunch 10/15: Josh Alman on Laurent Phenomenon Sequences

Dan Michael Stubbs dstubbs at stanford.edu
Tue Oct 13 11:43:55 PDT 2015

```As is our Thursday custom, there will be a theory lunch with food at 12:15pm and theory at 12:30pm in room 463A, with speaker Josh Alman.

Hope to see you there!

-dan
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Title: Laurent Phenomenon Sequences

Abstract: Consider the sequence of numbers defined by x_1=x_2=x_3=x_4=1, and then x_n = (x_{n-1}x_{n-3}+x_{n-2}^2)/(x_{n-4}) when n > 4. A priori, this should be a sequence of rational (non-integer) numbers, since we divide by a previous term in our recurrence. Nonetheless, if you keep computing bigger and bigger terms in the sequence, they all appear to be integers! The usual tricks to show a sequence consists of integers don't seem to work here, but instead a neat combinatorial notion called the Laurent phenomenon does the trick.

In this talk, I'll tell you about work done with Cesar Cuenca and Jiaoyang Huang, building off of results by Allan Fordy and Robert Marsh, investigating sequences of integers explained by the Laurent phenomenon. I'll present an algorithm which proves that a sequence has this property. I'll then discuss applications to physics and cryptography.

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