# [seqfan] Re: boxes in boxes

franktaw at netscape.net franktaw at netscape.net
Sat Sep 26 01:20:37 CEST 2009

```I'd skip the comment.

The correspondence is trivial.  Each of your boxes is a node of one of
the trees in the forest.  The label on the box is the number of points
associated with the node.  One node is a child of another if the first
corresponding box is contained in the second.

So you're just describing the same thing with different words.

-----Original Message-----
From: Jon Wild <wild at music.mcgill.ca>

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

```