2025-10-21 13:09:55: add post abt polya urn model

Author: xyzqm

content/posts/polya-urn.md

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+---
+title: The Pólya urn
+tags:
+  - combinatorics
+  - probability
+---
+
+The model is as follows ([Wikipedia](https://en.wikipedia.org/wiki/P%C3%B3lya_urn_model)): you have $n$ groups, each of which starts initially with 1 person. Every second, a new person joins a random group, where their probability of joining a given group is directly proportional to that group's size. For instance, if $n = 2$ and the groups currently have sizes $2$ and $1$ respectively, a new person will join the first group with probability $2/3$ and the second with probability $1/3$. Essentially, all the people are sheep and tend to follow the crowd. Now, the problem is the following: after $t$ seconds, what is the distribution of possible group sizes (i.e. what's the chance the $i$th group contains $a_i$ members)?
+
+> [!note]- Solution
+> It turns out that all possible group sizes are equally likely!
+> 
+> To see why, consider reframing the problem in the following way: there are initially $n$ people _in a line_ and $1$ divider between each of them, for $n - 1$ total. Then, every second, each new person randomly inserts themselves in front of a person already in line. Do you see how this model is equivalent?
+
+

content/thoughts/edits.md

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+---
+title: Edits
+---
+Ideas:
+- Heaven Can Wait/Timebomb
+- my heart so cold I think I'm down with ice
+
+[SUPERPOWERS|ITSV](https://www.youtube.com/watch?v=sXJRiDsUyF8)

content/thoughts/inverse-partition.md

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+---
+title: Inverse Minimum Partition
+---
+#dp #segment-tree #partition
+https://codeforces.com/contest/2159/problem/D2
+
+A very cool problem that relies on two clever observations
+
+Main lesson: it's always good to run/reason through a few test cases by hand and get a reasonable idea of what upper bounds are
+
+Also, when you make an observation, simplify it! 
+- i.e. consider only suffix minimums => can assume array is strictly increasing