# [seqfan] Re: "Stable" Partitions

Nacin, David NACIND at wpunj.edu
Sat May 9 04:51:52 CEST 2020

```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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