Fascinating Lattices Problem

Find the number of ways in which 20 α\alpha and 20 β\beta can be arranged in a row such that upto any point in the row, number of α\alpha is more than or equal to number of β\beta.

Note: You may use calculator, if required.

I posted this question one or two weeks ago (as a problem)! But got no response! So I thought of deleting it and reposting it here as a note. So that some real 'geniuses' can help me solve it out.

Note by Pranjal Jain
5 years ago

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This is a fascinating lattices problem. Here is a diagram that might help!

Imgur Imgur

Finn Hulse - 5 years ago

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@Pranjal Jain If you don't understand what I'm talking about, think of ordering the 40 characters like moving on a grid. A step north is like adding an α\alpha to the sequence, and a step east is like adding a β\beta.

Finn Hulse - 5 years ago

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1) Pic is not appearing 2) I tried modelling it that way but didnt reached answer.

Pranjal Jain - 5 years ago

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@Pranjal Jain Yeah... That's strange because I can't see it either. The link is https://imgur.com/5Nxc7RN.

Finn Hulse - 5 years ago

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@Pranjal Jain Also, I calculate the answer to be 6564120420.

Finn Hulse - 5 years ago

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@Finn Hulse Correct!! It comes out to be (2nn)n+1\frac{\dbinom{2n}{n}}{n+1} for n α\alphas and n β\betas.

Pranjal Jain - 5 years ago

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@Pranjal Jain Exactly! It's a Catalan number. :D

Finn Hulse - 5 years ago

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@Finn Hulse Eh!? You seem much more awesome and interesting than what I imagined!

Pranjal Jain - 5 years ago

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@Pranjal Jain Haha, thank you. Have you ever heard of Bertrand's ballot theorem? You NEED to check it out if you like this type of problem. It's one of the most beautiful theorems in my opinion. :D

Finn Hulse - 5 years ago

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@Finn Hulse Oh! Ill check it out after a sleep! Its 5:30AM in India. Lol! Going to sleep!

Pranjal Jain - 5 years ago

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@Finn Hulse Couldn't sleep!! I read complete wikipedia page of that Ballot theorem! You were correct... One of the most beautiful theorems! Any wiki page for it here on brilliant?

Pranjal Jain - 5 years ago

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@Pranjal Jain @Pranjal Jain No, a Wiki page does not exist for the Ballot Theorem as yet. Can you add one?

@Finn Hulse Note that to display an image, you have to link to the image file (typically ending with .png, .jpg .gif etc), as opposed to linking to an entire site.

Calvin Lin Staff - 5 years ago

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@Calvin Lin I have a camp at Kolkata. So i'll be busy next week. I will add one as soon as I reach home back!

Pranjal Jain - 5 years ago

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