The Most Important Sequence: The Catalan Numbers

This is our group submission for #some2

More about the Catalan Numbers:

Corrections:

At 6:43 the formula should only go up to 2n-1 instead of 2n+1

[chedzeesheeda]

The delivery of "Where do they come from? Dyck paths" is killing me and guaranteeing that this knowledge will stay in my head. I'm sure it wasn't intended for comedy, but my internet brainrot is so strong.

[derickd]

Really enjoying the format. I like that it is fairly fast, if I don't understand I just watch it a little again. It's also a nice combination of the 3B1B animation style but you also have your own face on the screen so it's like your lecturing. I think you'll be able to do a lot with this format

[johnchessant]

Great video! In fact, pi shows up in the Catalan numbers as well. Specifically,
pi = limit of 16^n / (n^3 (C_n)^2), as n -> inf

[BatteryAcid]

One of my favorite "wtf are you doing there" Catalan numbers moments is how it shows up in the coefficients of the mandelbrot set.

[ruroruro]

this video, but a vine boom plays every time he says "Dyck path"

[draziraphale]

This is really nice, thank you. I remember discovering a link between the Catalan numbers and Freundenthal's equation for root spaces years ago, it does crop up like pi!

[glenneric]

If only I could go back in time and tell Euler not to wear that newspaper on his head.

[Brainwizard.]

We finally reach a point where youtube contents are created by more and more professionals.
They surpassed already by such contents by far the university & school program.
Anyone who applied to a tech position will realize this over time. :)

[Mutual_Information]

Just discovered this channel. Excellent stuff. You’ll blow up I’m sure. Just don’t give up!

[Leo-iftn]

Hi, I am a brazillian student and I just wanted to say that I really enjoy your videos, hope you have sucess in this channel so more people watch and learn this content

[academicalisthenics]

I'm studying applied math and physics and I've encountered this sequence during my numerics class, the professor asked us to guess the next number in the sequence.

It's weird that so many of us never heard about these numbers, even though we had to complete a module on discrete mathematics beforehand.

[huhneat]

Going into someone's math notes and erasing all the parenthesis is the ultimate display of chaotic intent

[sstadnicki]

This was really nice! It took me a moment to realize that triangularizations were being counted in 'ordered' fashion — that is, the pentagon triangulates in five ways rather than just one, but once I got that the rest of that bijection clicked quickly. Your video flourishes and the soft humor in spots are really excellent.

[nimennacnamme]

4:05 That's a really NICE explanation! :)

[telotawa]

3:29 honestly, it's more obvious to me how to turn nested parentheses into dyck paths than it is for binary trees

[SimpleLivingHigherThinking]

This is the best video i have watched so far about catalan numbers i really understood the intuition behind catalan numbers and got the feel of i don't need to memorize the applications anymore....thanks a lot

[ashwinjain]

i already love this channel! combines my love of math and computer science

[WelshPortato]

New favourite Maths YouTube channel - really, really astounding videos. Hope you're keeping well x

[TheR]

1. How have I never learned (/recognized) these connections before? This is awesome!
2. Really like having the video of you as I feel like having a person looking at me helps me pay more attention (I remember there was some kind of study from a uni during the beginning of the corona lockdowns that attests that).

[robharwood]

Definitely would like to hear the story of the Associahedron! Hope you make a video on it. I've encountered Dyck paths -- and hence Catalan numbers and the related associativity -- many times in my hobbyist math wanderings, but never realized they could be depicted with some kind of geometric figure like the Associahedron appears to be. Fascinating! It also looks like it is probably connected with Group theory in some way!