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Combinatorics on a Linear Graph
By YouKn0wWho, 5 years ago, In English

Find the number of ways to select $$$k$$$ non-intersecting edges from a linear graph with $$$n$$$ nodes. In a linear graph every $$$i$$$ and $$$(i - 1)$$$ is connected by an undirected edge for each $$$i$$$ from $$$2$$$ to $$$n$$$.

Two edges intersect if they share some node. Two ways are different if there is some edge which is selected in one way but in the other.

Constraints: $$$1 \leq k < n \leq 10^5$$$

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5 years ago, # |
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$$$\binom{n-k}{k}$$$

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    5 years ago, # ^ |
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    Can you explain a bit, please?

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      5 years ago, # ^ |
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      nvm. I have just found it here. I am so dumb!

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      5 years ago, # ^ |
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      After selection you can contract the $$$k$$$ edges. Then there are $$$n-k$$$ vertices left, and we just have a size-$$$k$$$ subset of these $$$n-k$$$ vertices.

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    5 years ago, # ^ |
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    can you elaborate it??

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