Walking city streets: Catalan Closed Form (visual proof from lattice paths)

In this video, we show how to provide a closed form for the number of northeast lattice paths to the point (n,n) that don't pass below the line y=x. The number of such lattice paths is counted by the famous Catalan numbers.

For other videos discussing lattice paths (and those mentioned in this video), check out

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#catalannumbers #catalan​ #manim​ #math​ #mtbos​ ​ #animation​ #theorem​​ #visualproof​ #proof​ #iteachmath #mathematics #binomialcoefficients #latticepaths #discretemath #combinatorics #enumeration #closedform

This animation is based on the well-known interpretation of Catalan numbers as counting restricted lattice paths. I recommend the Wikipedia site or the books Enumerative Combinatorics I and II by Richard Stanley for someone wanting to know more about Catalan numbers.

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Music in this video:

[robnicolaides]

Lovely video! Is there a proof that corresponds to an algorithm that reflects the closed form formula more directly without using the subtraction of binomial coefficients?

[UDHAV]

I had a similar question in my math test once, the question was about the number of shortest paths a man can take to reach a specific intersection of a city on the north-east line in which the roads are laid out as grid lines on a plane without counting those paths that lie below the north-east. I had a similar thinking process where I thought that to reach a specific intersection (n, n) on the northeast line with the shortest path, the man must travel n north units and n east units, therefore the total number of shortest paths are the number of possible arrangements of n north units and n east units which are equal to (2n)!/(n!n!) and since we won't include those below the north-east line, we would have to remove half the approaches and therefore (2n!)/(2(n!n!)). Not sure if it's correct doe :)

[Daniel-mtxr]

not sure which one is the previous video you refer to? thanks

[hemadhumal]

From, Where did u get this content?.. 😅