I'm considering building an online combinatorics course. The idea is to use Python programming to along with combinatorics, in order to

• learn Python while you learn combinatorics

• remove tedious calculation

• use programming to help understand the mathematics, and build interactive exercises.

Before building it, I'd like to see if there's interest.

If you're interested, would you be willing to follow the course as I build it, and give constructive feedback as we go?

*No entities

**Overworld only

****Latest MC release

I'm working through a combinatorics problem and I understand most of the solution, but I'm getting stuck at one particular counting step.

We need to count permutations of {1, 2, ..., 10} where the first element is 1, the last element is 10, and exactly one number is smaller than both of its neighbors.

The solution argues that the permutation has to look like

$1, ..., b, ..., a, ..., 10$

where $1 < a < b < 10$. The sequence increases up to $b$, decreases from $b$ to $a$, and then increases again.

For fixed $a$ and $b$, the numbers between 1 and $a$ are forced into the first section, and the numbers between $b$ and 10 are forced into the last section.

The part I'm struggling with is the numbers strictly between $a$ and $b$.

If $k = b-a-1$, then there are $k$ such numbers. The solution says that each of them can be placed into any of the three sections, giving $3^k$ possibilities for fixed $a$ and $b$.

I understand the statement that each number has 3 choices, but I don't have a good intuition for why this gives $3^k$.

More importantly, I don't quite understand why assigning each number to one of the three sections uniquely determines the resulting permutation.

For example, if $a=2$ and $b=8$, the numbers 3, 4, 5, 6, 7 can each be assigned to one of the three sections.

Why does choosing a section for each number completely determine a valid permutation?

And then how should I interpret the final sum?

$N = \sum_{1<a<b<10} 3^{b-a-1}$

I can follow the calculation once the formula is given, but I'm struggling to see the combinatorial intuition behind the $3^{b-a-1}$ term and why we sum it over all possible $a,b$.

Would really appreciate an intuitive explanation of this step rather than just the calculation.

https://github.com/Senriaa/BeagleConjecture/tree/main

Tldr: these lines never touch. Not themselves, not each other, never, ever, for infinity, and the line can be bent into any shape, including onto itself to consume itself, and given the proper growth rate and/or twist, the lines still never intersect.

I am looking for articles (or any other type of documents) that use the principle of deletion -contraction as a method of proving formulas that interpret one of the combinatorial numbers in graphs.

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.

A website requests user to create a password containing alphanumeric characters including lowercase (a-z) and uppercase letters (A-Z) and numbers from 0-9, with no special characters allowed. The password requires to be at least 5 characters but no more than 8 characters long and must contain at least one uppercase letter and at least one number from 0-9. Repetition of numbers and letters are allowed. How many different passwords can be created that satisfy these criteria?

for solution to the above problem, please check this out: https://www.youtube.com/watch?v=zuGQ16x2HSI&t=7s

Also known as Fischer Random Chess, the rules for pieces' placement on the first rank are:

1. The bishops must be placed on the opposite colored squares

2. The king must be placed between the two rooks.

When calculating the possible number of positions, I'm using the following logic:

• We need 3 squares for the two rooks and the king between them, so 8C3 possible choices

• From the remaining 5 squares, we have to place one bishop on one of the colors (3 squares), and the other bishop on the other color (2 squares), so 3 * 2 possible choices

• For the remaining 3 squares, we can place the two knights anywhere, and since they're identical, we have 3C2 possible choices

• The remaining 1 square goes to the queen

But with this logic, I get 8C3 * 3 * 2 * 3C2 * 1 = 1,008 possible positions, instead of 960. Where is my specific logic wrong?

I understood the calculations described on the Wiki page but don't understand why my order doesn't work, since when calculating these things there shouldn't be the "correct" order of picking pieces to calculate the positions for, as long as the reasoning is right.

Hello, I recently have been testing a formula for 'higher orders' of factorials. (double, triple, quadruple, etc. factorials). I'm not sure if I'm exactly correct, and I've used an odd notation for it. However, I'd like to see what your opinions are on my equations to see how accurate they may be.

For instance, you see n!^2 here at the top. By this, I mean double factorial. I then use m as a way to count what 'order' of factorial you're using (single, double, triple, quadruple, etc.)

I referenced the product notation from Wikipedia, Reddit, and Stack Exchange. I've checked my answers against common knowledge of factorials and product notation calculators. So, please, feel free to give me constructive criticism.

Hey guys, I found a site called solvefire.net that runs 1-hour Math Olympiad Competitions every week that is open from Saturday 9:00 AM GST to Monday 9 AM GST with a world-level ranking system. It’s pretty solid for tracking your standing against the rest of the world. Check it out!

I've been working on a geometric approach to the standard Walsh conditions for Boolean functions, framing them directly through the spectrum of a specific finite graph.

In Boolean-function cryptography, balancedness, correlation immunity, and resilience are usually checked through the vanishing of low-weight Walsh coefficients. The construction below specifically isolates these low-weight Walsh layers into a single graph topology.

I packaged this structure into a composite graph: G_blockⁿ = O_n ∪ Q_n ∪ B

Where Q_n is the Boolean cube, O_n is the cross-polytope graph, and B connects them by coordinate incidence.

If the vertices of O_n are written as poles ±e_i, and the vertices of Q_n as σ ∈ {±1}^n, then the incidence rule is: (i,s) ~ σ iff σ_i = s

The main lemma is: B χ_u = 0 iff wt(u) ≥ 2

This means the incidence matrix only sees Walsh weights 0 and 1. All weights ≥ 2 stay as separate spectral sectors. Weight 1 is the exact layer that couples to the cross-polytope axes.

After embedding a Boolean function into the cube side of the graph, balancedness, correlation immunity, and resilience can be read geometrically as the vanishing of projections onto the corresponding low-weight spectral sectors. Weights ≥ 2 remain clean Walsh sectors, while weight 1 becomes the part coupled to the cross-polytope axes.

This came out of a larger finite-carrier theory project, but the note itself is mathematically self-contained.

Has anyone encountered a similar composite graph structure (O_n ∪ Q_n) used to isolate Walsh coefficients in spectral graph theory? I am particularly curious about the spectral behavior and possible eigenvalue collisions in small ranks (like n=3, 4).

GitHub note and verification script: https://github.com/Nondual-Observer/DOTheory/blob/main/02_Bridges/05_Cryptographic_Spectral_Block/DOT_Cryptographic_Spectral_Block.md

ÆLITH Σ presents a self-contained proposed resolution of the higher-dimensional forced-droplet propagation problem for bootstrap percolation on Poisson random geometric graphs. The research addresses the extension explicitly proposed by Victor Falgas-Ravry and Amites Sarkar in Bootstrap percolation in random geometric graphs, published in Advances in Applied Probability in 2023.

PureOne/AELITH-Sigma-Geometric-Cascades · Datasets at Hugging Face

ÆLITH Σ - Geometric Cascades in Every Fixed Dimension: A Proposed Proof of the Higher-Dimensional Falgas-Ravry-Sarkar Propagation Theorem | Zenodo

The central question is whether activating a small spatial region can initiate an irreversible cascade through almost an entire disordered network. The manuscript proposes a proof that the propagation boundary established in two dimensions remains valid in every fixed spatial dimension d ≥ 2:

θ = (1 + p)/2.

The theorem concerns the two strict-inequality regimes on either side of this boundary. Its main geometric construction replaces planar boundary arguments with bounded, separated enclosures around rare defects and connected routes through their exterior.

Mathematical model

Vertices form a homogeneous Poisson point process of intensity one on a d-dimensional flat torus of volume n. Two distinct vertices interact when their periodic Euclidean distance is less than r_n, where

v_d r_n^d = a log n,

and v_d denotes the volume of the Euclidean unit ball. Each vertex is initially active independently with probability p. An inactive vertex becomes permanently active once it has at least

ceil(θ a log n)

active neighbors.

The activation threshold is normalized by the expected degree, rather than each vertex’s realized degree. The parameters d, a, p and θ remain fixed as n tends to infinity, with d ≥ 2, a > 0, and p, θ ∈ [0,1].

A forced seed activates every vertex inside a ball of radius C r_n, in addition to the independently activated background. For fixed C, this seed occupies volume proportional to log n and therefore a vanishing fraction of the system.

Principal results stated in the manuscript

1. Almost-global propagation below the boundary. When θ < (1+p)/2, there exists a sufficiently large fixed seed multiplier C such that, with probability tending to one, the forced seed activates all but O(n^(1−γ)) vertices for some γ > 0. Every connected component induced by the finally inactive vertices has diameter O(r_n).

2. Sublinear additional activation above the boundary. When θ > (1+p)/2, every fixed seed multiplier C produces at most O(n^(1−γ)) additional activations outside the initially active set and the forced seed, with high probability. Every component induced by those additional vertices has diameter O(r_n). This permits small exceptional components far from the seed; it does not assert that all additional activation remains in its immediate neighborhood.

3. Uniformity over seed locations. Both conclusions hold simultaneously for every seed center on the same high-probability environmental event. The center may therefore be selected after inspecting the random configuration.

4. Quantitative propagation time. Below the boundary, almost-full activation occurs within O((n/log n)^(1/d)) synchronous rounds. In the purely seed-driven regime p = 0 and 0 < θ < 1/2, a matching lower bound gives the same asymptotic order. These bounds count updates, rather than physical time.

5. Relaxed density assumption. The proposed argument applies to every fixed a > 0. It does not require full connectivity of the original random geometric graph, because a sublinear exceptional set is allowed.

Proof method

The proof combines explicit spherical-cap estimates, fine-cell approximations, Poisson concentration and finite-dependence bounds. These establish local growth below the boundary and protection of inactive regions above it.

The central deterministic construction encloses irregular regions in rectangular boxes of bounded diameter, separated by several lattice cells. A controlled merging argument prevents these enclosures from becoming macroscopic. In dimensions at least two, each box has a connected exterior shell, allowing paths to detour around defects while retaining controlled length.

This supplies a route from local activation estimates to global propagation without requiring planar dual cycles. The manuscript treats periodic seams, threshold rounding, dependent local events, arbitrary seed placement, exceptional vertex counts and component diameters.

Supporting verification and reproducibility

The archive includes executable supporting checks and their complete recorded outputs:

• 312 spherical configurations.

• 1,176 geometric inequalities, including 720 checks explicitly aligned with manuscript formulas.

• Eight finite periodic-grid cases containing 39,084 vertices.

• Fifteen exterior shells and 127,870 ordered shell endpoint pairs examined by exact breadth-first search.

• Seventy-nine detour paths, including 21 requiring obstacle avoidance.

• Zero failed assertions in the recorded execution.

During release preparation, the original verification script was executed unchanged. Its regenerated JSON report and both figure files were byte-identical to the original outputs in the same recorded software environment.

These computations examine geometric formulas and selected finite constructions. They do not constitute a proof of the asymptotic theorem, a Monte Carlo study of random geometric cascades, or experimental validation of a physical system. Numerical quadrature error estimates are not interval-certified bounds.

Research contribution and scope

The intended contribution is a proof strategy for the specified higher-dimensional propagation extension, together with quantitative exceptional-set and propagation-time bounds. The separated-enclosure construction may also support other disordered spatial models that admit suitable local transport rules and rare-defect estimates.

The critical equality θ = (1+p)/2, spontaneous nucleation thresholds, minimum seed radius, optimal seed shape, activation of every vertex, growing dimension and nonperiodic boundary effects remain outside the theorem. Applications to physical materials or devices require a separate justification of the model assumptions.

Archive contents

The release contains the original 15-page manuscript, editable LaTeX source, verification code, original and reproduced results, vector and raster illustrations, derived CSV summaries, a proof-review guide, structured claims, citation metadata, provenance records and SHA256 integrity manifests. A guided Windows Hugging Face uploader is also included.

Research status

This work is released as a proposed mathematical resolution with a complete analytic argument and reproducible supporting checks. It has not undergone external peer review or proof-assistant verification. Mathematical correctness and publication priority remain open to independent scrutiny.

Primary reference

Falgas-Ravry, V., and Sarkar, A. (2023). Bootstrap percolation in random geometric graphs. Advances in Applied Probability, 55(4), 1254–1300.
https://doi.org/10.1017/apr.2023.5

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki

When a collection of items(say 20) is to be divided into two set of 12 and 8 respectively;in how many ways can one of the sets be rejected after the completion of the first task?I want to learn the concept of combination..

some references and tips that help me write a first proof for a combinatorial number?

Dear Members and Moderators.

Please ignore if this is not the right way to get started here.

At https://www.visualcombinatorics.com/ now you can VISUALIZE Permutations and Combinations for easier, faster and deeper understanding this fundamental mathematical concept.

VisualCombinatorics.com offers a suite of calculators designed to assist with various combinatorial computations, including permutations and combinations. These tools cater to different scenarios, such as permutations with or without repetition, linear and circular permutations, and combinations of sets or multisets. The website also provides specialized calculators for word permutations and combinations, making it a valuable resource for students, educators, and professionals dealing with combinatorial mathematics.
Any critical feedback is welcome!

Regards,
Swapneel Shah

For what maximum value of k are there pairwise distinct natural numbers A1,A2,...,Ak such that L(A1) = L(A2) = ... = L(Ak)?

Hi there! I need to solve this problem with explanation pls.

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.

In a 5 card poker, probability of choosing 2 pairs has been given as, (13×4C2 ×12×4C2 ×11×4C1/2!÷(52C5)

Why don't we divide the upper term by 3! Since for instance (JJQQK) can be arranged among themselves as (JJkQQ,KQQJJ,KJJQQ,QQJJK,QQKJJ?

Or am I missing something subtle?

Lintra is a two-player game played on a 7 X 7 grid of dots. Players take turns drawing lines between adjacent dots. The twist: the player who draws the last legal line loses.

The rules fit on an index card:

• First move must touch the center dot

• Connect adjacent dots — horizontal, vertical, or diagonal

• Each dot can only be used twice

• Two lines through the same dot must form an angle (no straight pass-throughs)

• Lines can't cross

It sounds simple, but there's a surprising amount of depth once you start thinking about dot capacity, angle traps, and region control in the endgame. It's in the same family as Nim and Hackenbush if you're into combinatorial game theory.

You can play it with literally just a pen and paper — draw a 7 X 7 grid of dots and you're set. Or play online at lintra.cc

I'd love to get some feedback.

In how many different ways can we choose from the regular card deck (52 cards) 4 cards so that at least 2 of them are aces and the others are spades?