Many combinatorial identities can be proved by simply telling an intuitive story. The story doesn’t end there! When generalising existing theorems, a surprising pattern often appears, called the ‘q-analog’ of theorems. And what does all these have to do with boxes and symmetric functions? It all starts with a curious question at a trivia night…

As we head down to the pub for a long-awaited trivia night since the lift of restrictions, the canonical questions to ask are: ‘what trivia rounds should we include?’ and—being the mathematically-minded young people that we are—’how many ways can we choose the trivia topics?’

Say that we have 10 candidate topics from which we’d like to pick 6 for tonight. Assume all rounds are different, and people don’t mind the order of the rounds. Then, we usually denote the total number of ways to select as  .

One fascinating thing about binomial coefficients  is that they have many identities that can be proved by simply counting the same thing in two different ways. For instance, Pascal’s rule

As we head down to the pub for a long-awaited trivia night since the lift of restrictions, the canonical questions to ask are: ‘what trivia rounds should we include?’ and—being the mathematically-minded young people that we are—’how many ways can we choose the trivia topics?’

Say that we have 10 candidate topics from which we’d like to pick 6 for tonight. Assume all rounds are different, and people don’t mind the order of the rounds. Then, we usually denote the total number of ways to select as  .

One fascinating thing about binomial coefficients  is that they have many identities that can be proved by simply counting the same thing in two different ways. For instance, Pascal’s rule

Yifan Guo
The University of Melbourne  View Yifan Guo
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