# [seqfan] Re: "Stable" Partitions

Sun May 10 02:15:35 CEST 2020

``` Dear Luca,
Thank you very much for pointing A076864 and A191646. They are fascinating, but I think the image you shared for A191646 is different than what I talked about. I need to study these two sequences and their references. There is a rich literature there and I am happy I found it.
Dear Dr. Nacin,
You are right. My terminology is confusing. For 8, I didn't consider 2//3/2/1 a "full" cycle because 1 is on the outside. The same thing with these two partitions of 10 https://justpaste.it/2dgr4.

I think another way to look at this issue is in matter of "bonds". Each even number n has n/2 bonds. What we need to do is to see how these bonds are connected. For example, 10 has 5 bonds. We find all the combinations of these five bonds.
5/////51/5////4etc.

Best,
Ali

On Saturday, May 9, 2020, 6:01:42 PM EDT, Nacin, David <nacind at wpunj.edu> wrote:

Luca,       I found that molecule, but discarded it since I don't think he's counting that one. My thought is that his cycles have to have at least 3 elements.  This is the reason that he has an answer of 2 cycles from the partitions of 8.  (Otherwise, he would have to count the one with two 4's and four bonds between them.)
-DavidFrom: Luca Petrone <luca.petrone at libero.it>
Sent: Saturday, May 9, 2020 3:20 PM
To: Nacin, David <NACIND at wpunj.edu>; Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: Ali Sada <pemd70 at yahoo.com>
Subject: Re: [seqfan] Re: "Stable" Partitions Dear David,
if this is what Ali was meaning, the sequence is A076864, the number of connected multigraphs with n/2 edges. Here is the picture for n=10:
https://oeis.org/A191646/a191646.png
Best,
Luca

Il 9 maggio 2020 alle 20.03 "Nacin, David" <NACIND at wpunj.edu> ha scritto:

Ali,
Thank you for nice explanation.  I think I understand now.  For 10 you would get the ones in the molecules1 picture?  I noticed that some partitions have multiple configurations as you get larger, for example, the one shown in molecules2.  Thus you could either count the number of partitions which can be turned into one of these cycle diagrams, or the total number of diagrams for each n.  I haven't looked at either in any detail, but they are really fun questions so thanks for sharing!
-David

From: Luca Petrone <luca.petrone at libero.it>
Sent: Saturday, May 9, 2020 9:37 AM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>; Nacin, David <NACIND at wpunj.edu>
Cc: Ali Sada <pemd70 at yahoo.com>
Subject: Re: [seqfan] Re: "Stable" Partitions
Dear Ali,
the sequence is interesting from a chemical point of view. As far as I understood it is the number of "ideal" molecules having exactly n bonds. My humble suggestion would be to count only the configurations for which you have only one molecule (if you have more than one, each of them is already included before).
Best Regards,
Luca
> Il 9 maggio 2020 alle 5.33 Ali Sada via SeqFan <seqfan at list.seqfan.eu> ha
> scritto:
>
>
>  Hi Dr. Nacin,
>
> Thank you very much for your email. I really appreciate it. And sorry again if
> my language and notation knowledge is not sufficient. I will borrow Olivier’s
> notation and use “/” to show the bond.
>
> You are right. The number of bonds attached to each part are equal to the
> part. The parts are allowed to be connected by more than one bond, e.g.
> (3///3).
> Some partition could be stable in more than one way. For odd numbers, there
> are no stable partitions. Even numbers have stable and unstable partitions.
>
> The first stable "molecule" is  1/1, which is a partition of 2.
>
>
> The first cycle appears in the partitions of 6:
>
> (1/1) (1/1) (1/1) is stable with three equal molecules.
>
> (1/1)(1/2/1) is stable with 2 different molecules.
>
> (1/2/2/1) is stable. This partition could also be stable in this form (1/1)
> (2//2).
>
> (2+2+2) is stable (a cycle where each 2 is connected to the other two 2’s by
> one bond.)
>
> (3+1+1+1) is stable (3 in the center connects to each 1 by one bond.)
>
> 1/3//2 is stable
>
> 3///3 is stable
>
> The partitions (6), (5+1), (4+1+1) and (4+2) are unstable.
>
>
> As for 8, below are some of its stable partitions:
> 4////4
>
> 4///3/1
>
> 2//4//2
>
>      1
>     /
> 1/4/1
>   / 1
> 3//3   (a cycle)
>  \ /
>   2
>
> 1/2/3//2
>
> 1/2/3/1
>      /
>     1
>   2/2   (a cycle)
>   /  |
> 2 /  2
>
> Etc.
>
>
> In general, we can say that all partitions of an even number with largest
> parts <= n/2 have stable forms. Partitions that have parts > n/2 have no
> stable forms.
>
>
> Best,
> Ali
>
>
>
>     On Friday, May 8, 2020, 10:52:02 PM EDT, Nacin, David <nacind at wpunj.edu>
> wrote:
>
>  Hi Ali,
>    First, I just wanted to thank you for the explanation you gave of the nine
> kings operation on the three-by-three square.  That's an interesting setup
> with lots of nice mathematical questions, and you made it very clear.  I have
> cycles?  The number 8 has many partitions, so what is the criteria for
> entering one of these cycles?  I see with the two you picked you get
> 2-2| |2-2
>
> and
>  2/ \3=3
> I'm thinking for these examples that you want the numbers to be in a circle
> and that the number of bonds attached to each number to be equal to the
> number?  Is this correct?
> -David
>
> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Ali Sada via SeqFan
> <seqfan at list.seqfan.eu>
> Sent: Wednesday, May 6, 2020 7:45 PM
> To: Sequence Fanatics Discussion List <seqfan at list.seqfan.eu>
> Cc: Ali Sada <pemd70 at yahoo.com>
> Subject: [seqfan] "Stable" Partitions Hi Everyone,
>
> I am not familiar with the literature of partitions, and I would really
> appreciate it if you could show me the OEIS sequences that are associated with
> the idea below.
>
>
> k is a part of partition of a positive integer n. k has k number of “bonds”
> that connects it to the other parts of the partition. For a partition to be
> “stable” all the bonds of its parts must be fulfilled.
>
> Some partitions form a cycle or cycles. For example, 8 has 2 cycles. The first
> one is four 2’s each one connected the other by one bond. The second one is
> two 3’s each connected to each other by two bonds, and each connected to 2 by
> one bond.
>
> The sequences associated with this idea could be: Number of stable forms of
> partitions of 2n, number of distinct "molecules", number of cycles, etc.
>
> Best,
>
>
> Ali
>
>
>
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