[seqfan] Re: "Stable" Partitions

[Ali Sada]

 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 some questions about this one.  Which partitions are you using to form 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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