The Quantum Factorial is upon us.

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  1. My guess this is some q-analog stuff.
    There's a "q-analog" to a lot of functions from combinatorics, which are continuous, and the regular ones are the limiting case as q goes to 1.
    It's also related to "quantum" groups, and possibly quantum mechanics.

  2. To simplify the calculations at 10:53, we can notice that the quantum factorial is also recursive like the “normal” factorial, where [n]!_q = (1-q^n)/(1-q) * [n-1]!_q instead of multiplying everything from scratch.

  3. Michael, thank you for making the "click bait" comment up front! We all know how the YouTube algorithm works, but it's so refreshing to see a presenter call it out so candidly! 👍

  4. One easy way to count the inversions is to go through the numbers in the 1st, 2nd,…,nth place, counting how many numbers before them are larger than the number in the current place.

    for example, for 4 2 1 3:

    4 –> 0 larger
    2 –> 1 larger
    1 –> 2 larger
    3 –> 1 larger
    total: 4 inversions

    This also proves the formula for the max # of inversions, n*(n-1)/2, as follows: you can at max have 1 inverted (larger) element before the 2nd, 2 before the 3rd, … (n-1) before the n-th, thus in total:

    1+2+…+(n-1) = n*(n-1)/2 inversions.

  5. One thing I truly appreciate about these videos (beyond the interesting mathematics) is the legible consistent handwriting and the organization of spacing, coloring, and symbols. The consistent speech patterns, delivery, and general organization of thoughts are clear as well. It's not very often I cannot follow something.

  6. If it counts the „number of swaps“ then it could also be used to determine the sign in the sum of a determinant, which is exactly the number of swaps counted and positive if it is even and negative if it is odd, right? Or how could we relate this to matrices, I mean a determinant is closely related to permutations and in my first semester linear algebra course we were given the definition with the permutation formular.

  7. Nice! On my combinatorics course I've learned a lot of finite and infinite formal power series and what sequences they generate. This is a very interesting topic!

  8. Is there an advantage to writing the q-factorial using the 'sum of finite geometric series' formula for this discussion? In this explanation, Michael is only using the polynomial expansion anyway, so (for this application, at least) why not just use that as the definition?

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