Catalan Numbers - Numberphile 

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Sophie's enthusiasm about the Catalan's number and after that about Pascal's triangle makes the whole episode so fun! :)

I never thought I would say this, but here it goes: "I want a Pascal's triangle talk".

BTW that XY interpretations restrictions are easily recognized as brackets (). There are always an equal number of open and close brackets, but there can never appear more close brackets than open ones that have already appeared in the sequence

Lol when she said, " I gotta get this right" 🤣 Deeeck words.

Sophie's contributions to this channel are starting to become addictive, partly for the content but mostly for her enthusiasm.

Pascal's triangle is the ninja behind the curtain of mathematics. It's everywhere, always jumping in the surprise me.

Would have been funny to end it with cameraman slowly walking out the door and we could still faintly hear Sophie talking with enthusiasm

The binary trees shown for C_5=14 contain two duplicates (corresponding to Dyck words XXXYYXYY & XYXXYYXY) in the thumbnail (3rd row) and at 2:40 (3rd row) & 11:20 (far right). Missing are those corresponding to Dyck words XXXYXYYY & XYXYXXYY.

The paper change interlude was much longer than the actual paper change this time! Lol 😂

There is a book called "Catalan Numbers" by Richard P. Stanley that lists 200+ different sets that give the Catalan numbers, mind-blowing...

What really blows my mind in math is when seemingly random, distant concepts turn out to be connected. Like Pascal's triangle at the end, or how pi shows up at ridiculous places.

I have been obsessed with Catalan numbers since 2013 when a professor showed how they related to binary trees. Seeing someone share this very particular obsession so strongly warmed my heart ❤

The method used in the Binary Trees in order to transform the problem is actually called Pre-Order Transversal of a Binary Tree. It's a way of unwrapping the binary tree, there's also In-Order and Post-Order. If you have a Binary Search Tree (that is, the binary tree is sorted in some way), then these different ways to transverse the tree give you a different meaningful result! So cool!

Found the video trying to learn about the catalan chess opening but stayed all the way thru. Well done! This was very interesting and fun👏

I love Sophie’s passion, and I love that she is willing to share it!

Combinatorics for the win! Great video, brilliant presentation and enthusiasm from Sophie. 😀

Every time I hear something shows up in pascal's triangle my immediate response is just "motherfu-". It feels like absolute magic

Thank you for the great explanation. As a contribution, ratio of two consecutive Catalan Numbers C(n)/C(n+1) converges to 1/4 when n grows since it equals to (n+1)(n+2) / (2n+1)(2n+2).

I want to thank your channel! Years ago when I was in high school I started watching your videos .I was terrible at math but your videos were so accessible and interesting that it helped inspire me to study more. I am now pursuing my master's degree in mathematics and I want to thank you!

*A filming tip* if you don't mind: Filming from the left side of a person drawing with their right hand would be better, so that the drawing hand doesn't cover the picture 🙃

Catalan Numbers me, a language nerd: "is it about numerals in the Catalan language?" 😂

The equivalence proof starting in 5:09 between pentagrams and trees is somehow sketchy and not very precise. If i take any graph created from joining fields in a pentagon i can choose any of the three blue nodes and say "this is the root" and uncoil it in such a way that that this one will be the actual root (topmost node). You can even make different binary trees choosing the same root. That doesn't show the 1 : 1 corespondance. At 5:40 you can even see that 4 of 5 are isomorhpic (ignoring chirality) and somehow each produces different binary tree. Unless the uncoiling procedure is more strictly defined than in here, this proof is not enough to say that those sequences are equal. Beside that, absolutely great video :)

I actually "discovered" these myself a while ago. I had a project where I needed a Python program to enumerate all the binary trees of a given size (even though I didn't know that's what I was doing). I saw the sequence 1, 2, 5, 14, 42 and thought "huh, interesting", and looked it up on the OEIS. To my surprise it was one of the longest entries there!

Any video starring Sophie is an instant gem. She's such a wonderful explainer.

Make more videos Sophie Maclean. Love your topics and energy!

The greatest talk on Catalan numbers I never though I would watch. The enthusiasm makes the show.

Such a nice thing to see a bright young woman like her. so excited about mathematics. Thank you so much!!!

Her energy is just so contagious! I can't get enough.

Catalan numbers crop up in some surprising places in computer science, such as range minimum queries. In a list of numbers, repeatedly find the smallest number in some range. You can do it, after linear time preprocessing, in constant time per query?! Part of the trick is to precompute cartesian trees for logarithmically-sized blocks, which is fast enough because the catalan number and the log cancel out.

Love it! Old school Numberphile!

Her enthusiasm about the Pascal's Triangle's relation made this episode SO MUCH better!

I adore Sophie’s energy !! 💜💜💜💜

A couple of years ago, while I was laid up recuperating from a heart attack, I had a friend bring me a calculator so that I could play with numbers. After a good while I discovered two completely different methods for determining the numbers in the diagonals of Pascal's Triangle. I'm not a mathematician, but I enjoy playing with numbers occasionally.

Would love to see more from her. Great pace and amazing at taking our hands through the conclusion 🎉 Bravo 👏

Wow I was blown away by her enthusiasm, contagious!

I just love this level of enthusiasm when geeking out over math explanations and seemingly weird interconnections!

I remember encountering Catalan numbers in university where they were presented as the number of ways to write balanced sets of brackets (essentially the Dyck words but with “(“ and “)” instead of “x” and “y”), but also as the number of ways to fill a 2xn grid of squares with the numbers 1 to n so that the numbers are increasing both left-to-right and top-to-bottom.

This is such a great way to visualize these fantastic numbers!

Please more things like this! this video was one of the most interesting Numberphile videos!

Phenomenal video as always!

I think Dyck words are a lot easier to understand if you use brackets; balancing brackets is much more intuitive than counting Xs and Ys with this weird abstract "Y's can't exceed X's'" rule. :)

Love the enthusiasm!

I am convinced that the academical field of mathematicians was brought into existence by puzzle-geeks, who loved puzzling so much, they found puzzles so obscure so that no-one of the rest of us even understands the puzzle any longer and then convinced the rest of us that it is crucial for the well-being of humanity that we pay them for locking themselves into a small chamber to solve said puzzle. Chapeau, puzzle-geeks, chapeau.

man, i love sophie, its so rare to see people like her that are close to my age and so passionate about maths :)

Catalan numbers are basically scaled central binomial coefficients. Binomial coefficients can be computed efficiently using a simple iteration rather than the usual n choose k = n-1 choose k-1 + n-1 choose k double recursion. This is what the programming language Python does behind the scenes. Fun fact: You can easily extend the Catalan numbers to C using the Gamma function rather than factorial. Catalan(z) = 1/(z+1) Gamma(2 z + 2) / Gamma(z + 1). The +2 and +1 come from the annoying definition of Gamma(n) as (n-1)! for natural n. def choose(N, k): def _choose_iterative(N, k): numerator = N denominator = k while k > 1: N -= 1 k -= 1 numerator *= N denominator *= k return numerator // denominator # guaranteed to be a whole number # preconditions and range reduction before jumping into the iteration assert isinstance(N, int) assert isinstance(k, int) if N N: return 0 elif k + k > N: # reflection formula k = N - k return _choose_iterative(N, k)

There is one more, indeed the most important step that needs to be included: Encouraging and supportive science education that fosters a joyful curiosity and a kind disposition to share.

I used to catalan numbers to iterate through all possible algorithms. Nice to see you guys making a video on them!

I could sit here and listen for an hour to Sophie talking about Catalan numbers

"This is where it becomes cool" --- indeed. The excitement showed is encaptivating. I was all in. - Cheers.

Best video in a while. Love the enthusiasm!!

10:10 -- "and this is where it becomes cool" - it was already so cool when I saw those three equivalences, but now I knew it was going to get way cooler!

I'll happily listen to her talk for an hour about something she's passionate about like this.

This is a really cool pattern to explore. Great job!

Sophie going all goofy at the end is the best thing I’ll see all day.

"Dyck is pronounced with a long *i* sound in the middle" This somehow reminds me of the "it's a bent finger" incident in the Egyptian fraction video lol

It's funny. During my PhD, I tried to count all the possible ways to find all the parametrizations of Feynman diagrams. If you know how to do that, then the relation to binary trees is straight forward, so I wanted to figure out all the binary trees with n leaves and get different ways to look at that. It was crucial to find the best possible parametrization and I was hoping that a different way to look at them would give some insight why some works and others don't. In the end it was a mostly futile attempt and I was stuck with trial and error, but I did actually find all the different representations that are given in this video (and one more, basically the number of ways to stretch a simplex into an n dimensional cube) and the formula, but I did not know they were called Catalan numbers :D It would have been very nice to know the name. Also, I definitely want a Pascals triangle talk.

About one minute into the video, when I saw 1 1 2 5 14 I went immediately to the "On-Line Encyclopedia of Integer Sequences". Anyone interested in combinatorics should know about this resource. The Catalan number sequence entry begins by noting that in all the tens of thousands of integer sequences it lists, the Catalan number sequence has the most entries.

Sophie is an awesome, awesome nerd. More of her, please.

Such infectious enthusiasm!

Another way to show the number of Dyck words of length 2n is comb(2n, n) - comb(2n, n - 1) is to consider a modified Pascal's triangle. In this modified triangle, the numbers are still the sum of the two above them, but the boundary changes (see below). Any path descending to a number within Pascal's triangle can be thought of as a word (not necessarily a Dyck word) where one descends diagonally left for an X and diagonally right for a Y. The Dyck condition would hold if the path never went to the right of the centre column and also ended on the central column. This can be realised by modifying the triangle so that the left diagonal is still all 1s but the column directly to the right of the central column is all 0s (so no path can go through them). If we take the triangle and move it one number to the right (so the apex is at (0, 1) instead of (0, 0), then negate all entries, the additive property still holds. Now adding this translated and negated triangle to the original triangle will form the required modified Pascal's triangle. The result follows immediately.

I'm gonna be drawing lines through pentagons today, thank you Sophie!

ERROR: The third row graphic of 5th order binary trees at 13:15 has a repeat of only 2 symmetries.

Sophie`s level of enthusiam at the conclusion on Pascal Triangle is like Liverpool just scored a goal

Wow, your explanation of Catalan Numbers in this video is absolutely fantastic! 👏 As a competitive programmer, I stumbled upon these gems a few months ago, and they've become an indispensable tool in my problem-solving arsenal. We use Catalan Numbers for various challenges, such as 1) Finding possible Binary Search Trees. 2) Generating Parenthesis Combinations. 3) Determining Triangulations on an N-gon. 4) Calculating possible paths in a matrix. 5) Dividing a circle into n chords. 6) Handling Dyck Words. 7) Navigating through Lattice Paths. Your breakdown has not only enhanced my understanding but also reinforced the significance of Catalan Numbers. Thanks for such a clear and insightful presentation! 🚀

Sophie Maclean may be my new favorite presenter. Very cool.

So cool, thank you for yet another great video

I love how she explains it

That was great! Thanks. Love learning.

These constructive set equivalence proofs are what I loved about theoretical computer science class.

Her abstract relations aptitude is off the charts.

I think using ( and ) rather than x and y is the better representation of Dyck words. It's just more obvious to people what the restriction means. And the tree-to-word translation is that they are both different ways to count the number of ways to associate n successive applications of a binary operator. I am particularly fond personally of the recursive formula for the Catalan numbers, that each Catalan number is the convolution of all the Catalan numbers that come before it. I still remember the first time I proved it. Spelled out, the recursive formula yields C1 = C0 C2 = C0 C1 + C1 C0 C3 = C0 C2 + C1 C1 + C2 C0 C4 = C0 C3 + C1 C2 + C2 C1 + C3 C0 . . .

I love the Catalan. I couldn't keep it under a quarter hour. We need more videos about all the cool things that go into Catalan numbers and how they are related. There is all sorts of things. Literally hundreds.

That was intense, thank you very much! 🙂

Lovely video! Another topic to scratch off my video list 😅

Watching videos like this for fun is why my friends call me a nerd 😅

I love the explanation and the energy! I also want a pascal triangle talk. Next numberphile video?

I came across Catalan numbers quite by accident when i was trying to calculate/simulate the distribution of "betting game lengths". Start with X>=1 and each game you bet 1. Each game has probably p of winning, with some payoff. I was interested in simulating how long could you bet on this game before you went bankrupt. Before too long i was deep into Wikipedia reading about Dyck words and binary tree traversal. Fun stuff for sure.

Thank you. Brilliant video👍👍

2:38 The picture is not correct as it only shows 12 different configuration ( in row 3 the two middle is the same and the two outer is the same ) The two missing configurations are pretty similar two the first two in row 1 with the exception that the shortest stem connects to the middle stem instead of the outer stem. So there are 14 configurations but not the 14 shown in this picture.

Great video, I love the enthusiasm! By the way, at 2:40, it looks like pattern numbers 10 and 11 (reading from left to right) are exactly the same as each other, as are numbers 9 and 12. I tried working these out myself, and I was wondering why I ended up with two patterns that I couldn't find in that picture. I guess 11 should have been a new unique pattern and 12 should have been its mirrored image but pattern 10 was accidentally copied into pattern 11 and then that was mirrored for pattern 12.

What a fantadtic presentation.

i really enjoy sophie and her attitude. more of her and prof tony

Symmetry seems important here for computer science. Thank you Sophie, you are great!

sophie is so interesting thank you sophie!

Geometric maths is always fascinating! Thinking outside the polygon!

Are you smiling at your screen now, like a kid? Don't worry, you are not alone 😀 I'll bookmark this to send to people when asked how can math be fun and exciting. Thank you for making it.

I wish everybody that one special person who looks at them like Sophie looks at catalan numbers and pascal triangle

For your binary trees of order 5: 9&12 are equivalent, and 10&11 are equivalent

The enthusiasm 👏👏

6:08 to 6:36 for anyone who didnt understand it, this alternate name may help. its Depth First Traversal/search (aka DFS)

Me, drunk on RU-vid at 2 am. Sees thumbnail of hexagons and numbers. "Oooh. A video on Catan numbers?" Me, as the hexagon question is proposed. "What does Settlers of Catan have to do with the Catalan numbers." Me, probably 75% of the way through the video. "I can't read."

I've been spoiled by Busy Beavers moving Trees of Graham's Numbers.

I was just searching this up yesterday and it appeared in my recommended.

Coming back to watch after I get home from work.

There's an error in the image thumbnail and at 2.40 . There are 2x two of the same binary tree on the second row

Now I want to see an episode with Sophie and Cliff Stoll together. I think the galaxy would explode

I'm a lil shocked you didn't show how Dyck numbers and binary trees relate to parenthesis in basic math equations.

Heres some more connections with Probability theory, specifically random matrices and free probability Consider a probability distribution with density 1/2π √(4-x^2). The moments of the distribution are 0,1,0,1,0,2,0,5,0,14,... Which means all the even moments are the catalan numbers. Compare to how the normal distribution has even moments as a double factorial (n-1)!!. Note that (n-1)!! Is the number of ways you can partition n points into n/2 pairs. Similarly, the catalan numbers are the number of ways you can partition n into non-crossing partitions. Consider a large random matrix (specifically a Gaussian ensemble). Its eigenvalues form a nice distribution. This distribution happens to be the semicircle distribution, which means the moments of the eigenvalues are the catalan nunbers. Those of you who have seen the central limit theorem will know that the sample average of iid scalar random variables converges in distribution to a normal. One way to prove this is to show the moments converge. It turns out if we consider independent random matrices (more specifically, freely independent random matrices) which are all distributed the same, it turns out sample average converges in distribution... ... To a semicircle. The moments converge to the catalan numbers. The proof is via using non-crossing partitions. The analogue of the central limit for random matrices is the semicircle distribution. You can read more about this in "Topics in random matrix theory" Tao and "Free probability and random matrices" Speicher, Mingo

Friend: "oh, look at this view!! Isn't it gorgeous?" Sophie: "yeah, that's nice. Have I told you about Pascal's Triangle?"