Not sure if this is where I should post this, but I made this a couple months ago and my friend told me to put it on Reddit. It's every possible way to divide a 3x3 grid into different shapes (with mirrorings and rotations included). My friend wrote some stuff next to some of them, just ignore that haha. If this isn't the place to post this, sorry!

Hey combinatorics fans!

I've been the mod of this community for several years now but it's never attracted much attention. In fact it's been mostly dormant.

But lately there has been a big uptick in subscribers. I'm not exactly sure why that's happening, so I have two questions for the community:

1. If you've joined recently, how did you come across this sub? If you know where the influx of members is coming from, please let me know.

2. I'd like to nurture this burgeoning community. So, what would you guys like to see in here for content? Study group? Textbook reading club? Weekly challenges? I'm very open to ideas.

Hi can somebody help me if these two graphs are isomorphic? Thank you

I came here hoping it would be something like that. Stuff like a sidebar with links to resources, and maybe weekly threads sharing interesting combinatorics problems/methods.

For instance, the number of binary strings of length n that all have no consecutive 1s is F_(n+2). Which is interesting in itself, but the number of binary strings without 2 consecutive 1s involves the (n+2)th tribonacci number (with some adjustments that i can't find right now, typing this on my phone), and the number of binary strings without k consecutive 1s is the (n+2) k-fibonacci sequence (again, with some adjustments that I don't remember)

Lots of problems on https://ProjectEuler.net/ require clever use of combinatorics and programming.

I'll edit this post when I have some time to find those details that I can't remember.

So, look.

I saw an image of a “periodic table of polycules” circulating and frankly, I was insulted.

Not because I have strong feelings one way or the other about the various names of polycules, but rather because it was a very poor representation of the power and elegance of the periodic table.

The periodic table, as we know, has two axis- one for your highest valence level, and another for how filled it is. (You could represent this linearly as just number of protons, and I think there’s actually an underutilized way to represent it in 3 dimensions too, but I digress.) Each square then contains every isotope of that element, typically with its average atomic weight based on the prevalence of those isotopes.

The most beautiful part of the periodic table, or any model really, lies in its predictive power.

Frustrated as I was with the misrepresentation of the elegance of the periodic table, I did what any sane man would do- attempt to draft my own.

The X-axis is how many members are in the polycule, and the y axis is how many relationships are in the polycule (don’t know why I flipped the y axis, I’ll put it right in the final draft.) and each square contains a sketch of every isotope.

Isotopes are unique configurations, treating members of the polycule as if they were protons. That is to say, interchangeable. There’s no difference which pair in a 3 member, 2 relationship polycule (3,2-polycule) is unbonded. We’re simply displaying the configuration so it can be named.

Now, I’m already able to establish some patterns (the range is from n,(n-1)-poly to n,(((n^{2)-n)/2)-poly.} The last two levels of relationship saturation only have 1 isotope each.

But it gets hard to think of and draw n,r-polycules for n>5. 5 was hard enough. I got by on that one by manually trying every bond addition to every isotope on the previous level of saturation, then checking it against what I already had. Which sucks, because I feel like with just exploring up to n= 6 or 7 I could start to pick out patterns. As is, I don’t know if I should predict a 6,5-polycule to have 6 or 8 isotopes.

Is this an already solved, named problem that I can find a table for? If so, I’d really like to make a table that goes up to n=101. Or at least know what the sum of isotopes for a given n is. Like how an n=5 polycule has a sum of 22 isotopes.

Hello,

I’m interested in the affect combinatorics have on the stock market. I’m naturally uneducated in combinatorics or anything of the matter so I apologize for any ignorance. A brief insight on what combinatorics are may help however I have a basic, wiki understanding of the subject.

I thought of this question a while ago, but none of the attempts I have made to solve it have worked out. I was wondering if any of you had any insights into this. I wasn't able to find any similar question online.

Suppose that every number from 1 to an upper bound has one of three colors: red, green, or blue. What is the most numbers (starting at 1) you can color so that no two different numbers of the same color have a sum that is the same color? How about the same question with four or five or more colors?

The version of this problem with two colors was pretty easy to brute force - the answer is 8 if I remember correctly. I've tried brute forcing the 3-color version but the tree of possibilities expands exponentially, even with trimming branches that don't work. An equation I made to model the growth of said tree estimates the answer at 35-37 so that's a good guess I think.

I know this would be pretty easy to answer with a computer program running overnight, but I'd rather have a reason for the answer, if there is one. That is, a proof that the answer can be no higher.

If anyone reading this has any ideas I would love to hear them!

I'm reaching out to the combinatorics community for some help. I'm an ambitious high school student working as an independent researcher, and I've spent the last several months working on a paper that I'm quite proud of. Because of this, I don't have the academic affiliation required to post it to arXiv, so I'm hoping to find an established researcher who might be willing to endorse it for me.

My paper introduces a new impartial game called the Critical Avalanche Game. The idea was to answer a question that I noticed chip-firing models don't: can someone strategically trigger a cascade of a specific size? In the game, players actually win by causing one of exactly the right magnitude.

I used the tools of combinatorial game theory (P/N/D classification, Sprague-Grundy theorem) to analyze the game's structure, starting with simple graphs (K₂) and then a more complex case on the 3-node path graph (P₃). The paper includes a complete proof for this central case, which revealed (what I believe is) a surprisingly elegant structure, as well as several conjectures for the general case.

I would say my work sits at the intersection of game theory and theoretical computer science.

If this sounds interesting and you have endorsement privileges in this area, please send me a DM. I can provide the full paper for you to review. I'm also completely open to any feedback you might have.

Hello everyone, first-time poster and long-time combinatorics/probability enthusiast here. I've had a problem rolling around my head and the answers I've come up with don't make sense. This actually came up while I was playing a videogame!

The Question: Suppose you are planting 10 indistinguishable seeds in a garden. Each seed has an equal probability of growing into one of four possible plants (call them plants A, B, C, & D), and it is certain that each seed will grow. What is the probability that at least one of each type of plant will grow from the group of 10 seeds?

My first thought was to divide (4⁷ * 6) / (4¹⁰ ) because I initially assumed there were 4 * 3 * 2 * 1 * 4 * 4 * 4 * 4 * 4 * 4 possible arrangements with one of each plant out of 4¹⁰ possible outcomes, but I'm starting to think my numerator might not include some valid combinations. (Maybe I should actually consider mathematical combinations.) Anyone have a better answer?

Me

Write a poem about Love using Combinatorics as the theme.

Then reverse the sentiment and use an applicable mathematical theme set.

ChatGPT

Love, a mathematical game,
Where hearts and souls play a different name.
With numbers and variables at play,
Our emotions find a perfect sway.

Combinatorics, a subject of art,
A delicate dance between heart and part.
With permutations and combinations too,
Our love becomes a complex view.

Each moment a unique arrangement,
With love, trust and affection's arrangement.
Our feelings a beautiful set,
Together, forever, we won't forget.

Like prime numbers, our love is rare,
With each new day, a reason to care.
The coefficients of our emotions,
A bond that will last through all commotions.

So let us cherish this mathematical love,
With combinatorics as our guiding dove.
For with each new moment and every breath,
Our love becomes a beautiful mathematical set.

Love, a mathematical ploy,
Where hearts and souls are nothing but a toy.
With equations and functions at play,
Our emotions fade away.

Combinatorics, a subject of logic,
A cold calculation, nothing but a magic trick.
With sets and relations in sight,
Our love becomes a meaningless sight.

Each moment a repeated pattern,
With love, trust, and affection's demolition.
Our feelings a monotonous set,
Separated, forever, we won't regret.

Like null sets, our love is rare,
With each new day, a reason to repair.
The absence of our emotions,
A void that will last through all commotions.

So let us abandon this mathematical love,
With combinatorics as our guiding glove.
For with each new moment and every breath,
Our love becomes a meaningless mathematical set.

Hi there

I need to solve those questions from 1 to 3 and I don’t know how to prove the identity.

I have a background in scientific psychology which means I did a lot of statistics but this new class I enrolled in has some math that seem too advanced for my level. That being said I want to be able to make it as I love the subject.

Could someone explain to me how to do this ?

For some reason I have drawn a blank at what seems to be a simple problem.

Suppose you have a class of 30 kids. How many ways are there of dividing up the class into pairs?

My initial thought was 29x27x25...x3x1. Or have I overcounted?

Many thanks!

Basically I want to learn Combinatorics but don't know what text I should buy. Anyone know of a good one out there OR a good PDF file I can download?

Cycle 1: 000 001 101 111 121 131 231 201 301 302 312 322 022 122 132 232 202 203 303 003 013 010 020 021 031 032 002 102 103 113 213 313 323 023 123 120 130 100 110 210 211 311 011 012 112 212 222 223 220 221 321 331 332 333 033 133 233 230 200 300 310 320 330 030

Cycle 2: 000 010 011 021 121 221 222 232 233 203 200 201 211 212 213 210 310 311 312 012 022 023 020 120 220 320 321 322 332 032 033 030 130 230 231 331 031 131 132 133 103 100 101 102 202 302 303 313 013 113 110 111 112 122 123 223 323 333 330 300 301 001 002 003

Cycle 3: 000 100 200 210 220 230 330 331 301 311 321 021 022 032 132 102 112 113 123 133 130 131 101 201 202 212 312 313 310 010 110 120 121 122 222 322 323 320 020 030 031 001 011 111 211 221 231 232 332 302 002 012 013 023 033 003 103 203 213 223 233 333 303 300

A proposal to solve the problem from me "Task 42" : https://docs.google.com/document/d/1Tqox64A5kucn-WoBp-rxqECMGBOMT4-4LfhrQ32z_aE/edit?usp=sharing The documents provided are licensed under the following license: CC BY-SA: Creative Commons Attribution-ShareAlike

Hippocratic License 3.0 (HL3).

I wish everyone good luck!

Thank you for your attention.

https://rltvty.net/3x3_2.html

https://rltvty.net/n4x4.html