Consider the integers $[1,n]=\{1,\dots,n\}$ and call subsets of the type $[a,b]=\{a,\dots,b\}$ with $1\le a < b\le n$ *intervals*. We say that two intervals $[a,b],[c,d]$ are *crossing* if either $a<c<b<d$ or $c<a<d<b$. Otherwise we say that the intervals are non-crossing. So $[1,3],[2,4]$ are crossing, but $[1,3],[2,3]$ or $[1,3],[3,4]$ are non-crossing.

Let $A_n$ denote the collection of pairwise non-crossing sets of intervals. So the first few are \begin{align} A_2&=\Bigl\{\emptyset,\{[1,2]\}\Bigr\}\\ A_3&=\Bigl\{ \emptyset,\{[1,2]\},\{[2,3]\},\{[1,2],[2,3]\},\\ &\qquad\{[1,3]\},\{[1,2],[1,3]\},\{[2,3],[1,3]\},\{[1,2],[1,3],[2,3]\} \Bigr\}\\ A_4 &= \Bigl\{ \emptyset,\{[1,2]\},\{[1,3],[3,4]\},\{[1,2],[2,4],[1,4]\},\dots \Bigr\}. \end{align}

It seems that there is an easy recursion for the number of such sets of non-crossing intervals: $$|A_n| = 2 \Bigl(2|A_{n-1}||A_2|-|A_2|^2|A_{n-2}|\Bigr)=8(|A_{n-1}| - |A_{n-2}|) \tag{*},$$ so that $$(|A_2|,|A_3|,|A_4|,\dots)=(2,8, 48, 320, 2176, 14848, 101376,\dots)$$ which seems to be https://oeis.org/A228568.

**Questions:**

- Is there some standard name for sets sets of non-crossing intervals in the literature? Is the number of such sets known?
- Is the recursion above correct?

My reasoning for (*) is as follows: Any non-crossing collection of sets on $[1,n]$ can be obtained from non-crossing collections on $[1,n-1]$ and $[n-1,n]$ or $[1,2]$ and $[2,n]$ by optionally adding the interval $[1,n]$. However, this is over-counting the intervals which can be obtained from combining non-crossing collections on $[1,2],[2,n-1],[n-1,n]$.

**Edit:** My reasoning for (*) was flawed and the conclusion was incorrect. The correct number of sets seems to be
$$(|A_2|,|A_3|,\dots)=(2, 8, 48, 352, 2880, 25216, 231168, \dots)$$
which is https://oeis.org/A054726.