# [seqfan] Re: Set of sets using minimal number of braces

franktaw at netscape.net franktaw at netscape.net
Sat Oct 29 15:21:33 CEST 2011

```I sent you a direct response when you emailed me directly; I don't know
why you didn't see it.

The one you are missing is:

{ { {} {{}} } }

-----Original Message-----
From: Aai <agroeneveld400 at gmail.com>

I do not know all the preconditions for a legal n-element set.
That said, can you show to me all 3 five-set-of-braces. I can only
think of two
sets:

{ {{{{}}}} }
{ {} {{{}}} }

Hallo franktaw at netscape.net, je schreef op 25-10-11 09:08:
> The number of sets that can be made with n braces is A004111. This
sequence
> has differences which are n repeated A004111(n) times. This means
that it is
> actually:
>
> 1, 2, 4, 7, 11, 15, 20, 25, 30, 36, ...
>
> More terms (billions of them) can easily be calculated using the
values in
> A004111.
>
> This sequence can also be described as the minimum number of nodes in
a rooted

> identity tree where the root has n children.
>
> You should submit it.
>
>
> -----Original Message-----
> From: Joshua Zucker <joshua.zucker at gmail.com>
>
> Hi folks,
> I was working with some 5th graders today on an introduction to set
> theory and we came up with a question whose answer doesn't appear to
> be in the OEIS.
>
> The question is, what's the minimal number of braces required to make
> an n-element set?
>
> For a zero-element set, {} is one set of braces.
> For a one-element set, {{}} is two sets of braces.
> For a two-element set, {{},{{}}} is four sets of braces.
> For a three-element set, {{},{{}},{{{}}}} is seven sets of braces.
> You might think from this pattern that we're on the usual 1, 2, 4, 7,
> 11, but then it's 15 next, not 16, because there are two different
> four-sets-of-braces objects you can put in, so the five-element set is
> {{}, {{}}, {{{}}}, {{{{}}}}, {{},{{}}}} and you save one set of
> braces.
>
> This leaves me pretty sure that the sequence is 1, 2, 4, 7, 11, 15,
> 20, 25, 31, 37, ... but I'm not sure how to create more terms.
> The first differences of this sequence are also interesting.
> And, the sequence of how many different sets can be produced with n
> sets of braces also looks good.
>
> Any ideas about how to automate the calculation of a bunch of terms of
> these things?  Or even better, a pointer to a known formula?
>
> Thanks,
> --Joshua Zucker
>
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--
Met vriendelijke groet,
@@i=Arie Groeneveld

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