# [seqfan] boxes in boxes

Jon Wild wild at music.mcgill.ca
Fri Sep 25 22:58:04 CEST 2009

```Dear seqfans,

Say you have a bunch of boxes, each labelled by an integer. Boxes can go
inside other boxes. The sequence I was just thinking about is the number
of ways I can give you a collection of boxes whose labels sum to n. For
n=3 for example, I can give you:

one box labelled 3;
one box labelled 2 and a separate box labelled 1;
a box labelled 2 with a box labelled 1 inside it;
a box labelled 1 with a box labelled 2 inside it;
three separate boxes labelled 1;
a box labelled 1 with another box labelled 1 inside it, plus a separate
box labelled 1;
a box labelled 1 with two separate boxes labelled 1 both inside it;
a box labelled 1 with a box labelled 1 inside it, with a third box
labelled 1 nested inside the second box.

That's 8 ways.

I checked by hand up to n=5 and got 1,3,8,24,71. Can someone confirm that
this sequence is A052855, "Number of forests of rooted trees of nonempty
sets with n points. (Each node is a set of 1 or more points.)"?

If so I'll add a comment, and if not I'll submit the sequence.

Jon

```