Hey there. I was looking around but could not find it: what is the number of chains in a “degenerate” interval of a poset? Thank you!

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Hey there. I was looking around but could not find it: what is the number of chains in a “degenerate” interval of a poset? Thank you!

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I assume that by “degenerate” interval you mean the trivial interval, namely , for $x$ in a poset. In this case, there is only one possible chain contained in , namely .

Sorry, I had a few typos in my reply. Below I repeat the second sentence:

In this case, there is only one possible chain contained in , namely

(Recall that a chain is a subset of a poset where any two elements in are related, or “comparable.”)

Got you,thank you! ðŸ˜‰