cp's OEIS Frontend

From Emeric Deutsch, Aug 21 2008: (Start)

Number of Dyck paths of height at least 4 and of semilength n. Example: a(5)=8 because we have UUUUUDDDDD, UUUUDUDDDD, UUUDUUDDDD, UUDUUUDDDD, UDUUUUDDDD and the reflection of the last three in a vertical axis.

Number of ordered trees of height at least 4 and having n edges. (End)

From Gus Wiseman, Jun 22 2019: (Start)

Also the number of non-crossing, capturing set partitions of {1..n}. A set partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting. The a(4) = 1 and a(5) = 8 non-crossing, capturing set partitions are:

{{1,4},{2,3}} {{1,2,5},{3,4}}

{{1,4,5},{2,3}}

{{1,5},{2,3,4}}

{{1},{2,5},{3,4}}

{{1,4},{2,3},{5}}

{{1,5},{2},{3,4}}

{{1,5},{2,3},{4}}

{{1,5},{2,4},{3}}

(End)