# [seqfan] Re: "Stable" Partitions

Thu May 7 12:06:31 CEST 2020

``` Dear Olivier,

Thank you very much for your response. I really appreciate it. Your example of CH4 is exactly what I am talking about. C has four bonds to fulfill, while H only has one. CH4 is stable because all the bonds are fulfilled. This is equivalent to the partition if 8 (4+1+1+1+1).

1 is allowed (like H in CH4), but it can only be on the “outside”.
To make things easier, let’s consider 2//2  as equivalent to /2/2\ . Cycles appear only when we have 3 or more parts.

Please check the picture in the link below. It explains everything I want to say. It is a partition of 22 (5+5+4+4+2+1). Each part fulfills its number of bonds. And of course this partition has other shapes.
https://justpaste.it/33ku4

Things get even more complicated in 3 dimensions. I think the least number that has a stable 3D form is 12 (four threes, each one on a vertex of a triangular pyramid).

And thank you for sharing the three sequences. Obviously I have a lot to learn!

Best,

Ali

On Thursday, May 7, 2020, 4:36:25 AM EDT, Olivier Gerard <olivier.gerard at gmail.com> wrote:

Dear Ali,

I am not sure what you describe is sufficiently well defined

There is a large literature on combinatorics inspired by the ideas of
molecules.

If I understand well your system, each number is an atom requiring a number
of bonds related to its size.

But in that case your examples are not completely consistent. You have to
specify more.

For instance : can 1 be connected to another part ?

Here is my first approximation (there are several missing, depending
on interpretation of the rules, ...)

If 1 is allowed

2:  1/1
4:  1/1 1/1 ;  2//2 ;  (+  /2/2\ (cycle))
6:  3///3 ; /3//3\  (cycle);  /2/2/2\ (cycle); 1/2/2/1 ; 2//2 1/1 ; 1/1
1/1 1/1;  2//3/1 ;  1/3/1 plus 1 over for a third bond ;
8:  /2/2/2/2\ (cycle) ; 4////4 ;  /4///4\  (cycle); //4//4\\ (cycle);
3///3 1/1  ;  /2/3//3/2\ (cycle)  ;  1/3//3/1 ; 1/4///3 ; 2//4//2 ;  1/4/1
(plus 1 over and below, like CH4)
(+ all couples of partitions of 4)
10: 5/////5  ;  /2/4///4/2\ (cycle) ; /2/3//3/2\  (cycle)  ; 4////4  1/1;
2//4//3/1  ;  1/2/3//3/1  (+ all partitions of 6 x all partitions of 4)

If 1 is not allowed

4:    2//2 ;  (+  /2/2\ (cycle))
6:    3///3 ; /3//3\  (cycle);  /2/2/2\ (cycle)
8:    2//2  2//2 ;  /2/2/2/2\ (cycle) ; 4////4 ; /4///4\  (cycle); //4//4\\
(cycle);  /2/3//3/2\ (cycle)  ;  2//4//2 ;
10:  5/////5  ;  /2/4///4/2\ (cycle) ; /2/3//3/2\  (cycle)    (+ all
partitions of 6 x all partitions of 4)

Thinking in partition terms, you might be interested in the following
sequences

A classical one is A047966
1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, ...
Number of partitions of n such that every part occurs with the same
multiplicity

It will list all cases where you can group parts evenly

two new ones are restriction of the previous sequence, I will create them
in the OEIS

0, 0, 0, 1, 1, 3, 2, 4, 5, 7, 6, 12, 9, 15, 17, 21, 20, 33, 28, 43, ...
Number of partitions of n such that every part occurs with the same
multiplicity
and with least part greater than 1 and more than one part.

0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 8, 9, 13, 15, 18, 20, 29, 28, 39, ...
Number of partitions of n such that every part occurs with the same
multiplicity
and with least part greater than 1 and more than one part and more than one
type of part

Olivier

On Thu, May 7, 2020 at 8:25 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
wrote:

> 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
>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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