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After some research, there seem to be some papers regarding the number of certain types of tunnels, but none studying their lengths, which is the parameter I would be interested in. Any other reference would be highly appreciated!

Update:

After some research, there seem to be some papers regarding the number of certain types of tunnels, but none studying their lengths, which is the parameter I would be interested in. Any other reference would be highly appreciated!

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Sam Hopkins
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Thank you very much for the comments. It turns out that what I am looking for is called either "chords" or "multitunnel" in FuckDyck paths depending on the author. It would be great to find a generating function involving their length. I will continue searching for references.

Thank you very much for the comments. It turns out that what I am looking for is called either "chords" or "multitunnel" in Fuck paths depending on the author. It would be great to find a generating function involving their length. I will continue searching for references.

Thank you very much for the comments. It turns out that what I am looking for is called either "chords" or "multitunnel" in Dyck paths depending on the author. It would be great to find a generating function involving their length. I will continue searching for references.

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Edit:

Thank you very much for the comments. It turns out that what I am looking for is called either "chords" or "multitunnel" in Fuck paths depending on the author. It would be great to find a generating function involving their length. I will continue searching for references.

Edit:

Thank you very much for the comments. It turns out that what I am looking for is called either "chords" or "multitunnel" in Fuck paths depending on the author. It would be great to find a generating function involving their length. I will continue searching for references.

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