A Surprising Combinatorics Bijection!

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In this video we count the number of partitions with m or fewer parts such that the largest part is at most n. We solve this by finding a bijection with a set of objects that are easy to count.
  1. Dr. Weselcouch
  2. Combinatorics
  3. Partitions
  4. Bijection
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Came for the great math. Stayed for the great shirt!

mattolbert
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This is really nice. So few students get exposed to these neat combinatorial ideas. Thanks, Dr. W!

routemath
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Your solution is very neat and I like it a lot, even though I'm not sure I understand a point. On the bot-left board you write that we are looking to solve the following question: "How many Young Diagrams fit in an n x m box?". However, you then look for every path that goes from (0, 0) to (n, m); didn't we loose the "fitting" part? With you proof, are you really counting the example λ = (4, 3, 3, 1, 1) when n = 10 (aren't you assuming that the (n, m) point is part of the path; shouldn't it be (k, m) with k <= n)?

JoachimFavre