Decomposing factorial of 300K as the product of 300K factors larger than 100K

http://gus-massa.blogspot.com/2025/04/decomposing-factorial-of-300k-as.html

A free book about Graph Theory and Additive Combinatorics.

https://yufeizhao.com/gtacbook/

A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra $A$ into its "basic parts". Formally, we say that $A$ is a subdirect product of $A_1$, $A_2$, ..., $A_n$ when $A$ is a subalgebra of the product
$$A_1\times A_2\times \cdots \times A_n$$
and for each index $1\le i\le n$ we have for the projection ${\pi }_i$ that ${\pi }_i(A)=A_i$. In other words, a subdirect product "uses each component completely", but may be smaller than the full product.

A trivial circumstance is that ${\pi }_i:A\to A_i$ is an isomorphism for some $i$. The remaining components would then be superfluous. If an algebra $A$ has the property than any way of representing it as a subdirect product is trivial in this sense, we say that $A$ is "subdirectly irreducible".

Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic $p$-groups ${\mathbb{Z}}_{p^n}$ and the Prüfer groups ${\mathbb{Z}}_{p^{\mathrm{\infty }}}$.

In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (https://math.chapman.edu/~jipsen/posets/si_lattices92.html) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?

We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", https://arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

For various (mathematical, meteorological, alimentary) reasons, I usually prefer 2π day.
Nevertheless, today I make the following offering:

http://arxiv.org/abs/2503.10002

Pjotr Buys, @Janvadehe and I used Shearer's induction to address the question:

How few independent sets can a triangle-free graph of average degree d have?

The answer is close to how many a random graph has.
What is perhaps surprising is just *how* close it comes.
(I queried the combinatorial hive mind about this last week.)

Your #DnD Party is Too Big
https://youtube.com/watch?v=0pc9Uf3vFDU&si=N2Dor4X31i93IGie

searching for any structures / theory that involve a particular operation on non-empty lists of postitive integers like "the length of the list multiplied by the least common multiple of all the items in the list"

any ideas? references to any literature would be very appreciated if you know of any.

1/2

I want to find a particular sequence of permutations. The requirements are:

* you start with the identity
* each permutation differs from the next by a permutation with only adjacent transpositions
* each number appears in every position at least once.

For example, for $n = 4$, here's an example:

1234,2143,2413,4231,4321,3412

What's the shortest such sequence as a function of $n$?

I have one algorithm that produces such a sequence that is length $2n +2$ for even $n$, and $2n-1$ for odd $n$. The attached photo should make the algorithm clear. But I don't have a good proof that this is optimal.

This is related to some #crochet I'm working on, surprisingly enough.

"How Anime Fans Stumbled upon a Mathematical Proof: When a fan of a cult anime series wanted to watch its episodes in every possible order, they asked a question that had perplexed combinatorial mathematicians for years."

#math #combinatorics #anime #4chan #compsci #computerscience

https://www.scientificamerican.com/article/the-surprisingly-difficult-mathematical-proof-that-anime-fans-helped-solve/

A question for the (combinatorial) hive mind.

There are a lot of extremal results that are matched asymptotically by some probabilistic construction, but with some gap, often quite substantial. I'm thinking about the Ramsey numbers R(k,k) or R(3,k), but examples of this phenomenon are prevalent.

I'm curious, does someone out there know of good examples of (extremal) results where some probabilistic construction (e.g. via a random graph) is matched asymptotically, and very precisely?

"Combinatorics in general and the topics of this work in particular have a special charm. It is characterized by the maximum simplicity of its basic concepts, since it basically operates only on finite sets. Also, its methods are usually modest, including some elements of number theory and the theory of finite groups and finite rings. Therefore, creativity in this field is very difficult. Its only basis is combinatorial fantasy, the kind of fantasy that produces the most surprising and valuable results in mathematics."

Jan Mycielski, commenting on work of Barbara Rowkowska (see https://arxiv.org/pdf/2502.06792)

Symbol of Combinatorics

Gottfried Wilhelm Leibniz, 1666

Starting out in mathematical research, especially in discrete mathematics, a big focus is problem-solving. It's like a race, and once you've solved one, you set out right away for the next adrenaline rush.

Take for granted a bustling market of open problems (again, especially in discrete mathematics). Scour papers or problem sites. Challenge close colleagues with the ones that eluded you. The harder, the better, right? There is occasionally awkward coffee talk of that intangible `taste' or `judgement', but, come on, less talk and more solving!

(please imagine here a subtly ironic tone in my voice)

(1/3)

#math
#graphtheory
#combinatorics
#ExtremalCombinatorics

A post of @11011110 has reminded me that (after a year and a half lurking here) it's never too late for me to toot and pin an intro here.

I am a Canadian mathematician in the Netherlands, and I have been based at the University of Amsterdam since 2022. I also have some rich and longstanding ties to the UK, France, and Japan.

My interests are somewhere in the nexus of Combinatorics, Probability, and Algorithms. Specifically, I like graph colouring, random graphs, and probabilistic/extremal combinatorics. I have an appreciation for randomised algorithms, graph structure theory, and discrete geometry.

Around 2020, I began taking a more active role in the community, especially in efforts towards improved fairness and openness in science. I am proud to be part of a team that founded the journal, Innovations in Graph Theory (https://igt.centre-mersenne.org/), that launched in 2023. (That is probably the main reason I joined mathstodon!) I have also been a coordinator since 2020 of the informal research network, A Sparse (Graphs) Coalition (https://sparse-graphs.mimuw.edu.pl/), devoted to online collaborative workshops. In 2024, I helped spearhead the MathOA Diamond Open Access Stimulus Fund (https://www.mathoa.org/diamond-open-access-stimulus-fund/).

Until now, my posts have mostly been about scientific publishing and combinatorics.

#introduction
#openscience
#diamondopenaccess
#scientificpublishing
#openaccess
#RemoteConferences
#combinatorics
#graphtheory
#ExtremalCombinatorics
#probability

I'm busy writing a #Python program that creates (virtual) #Lego #moc files for #LEOCad. But, I've hit a #combinatorics wall, and my #maths-fu fails me here...

Are/is there any #programmer and/or #mathematician who can help me?

See picture for what I'm trying to do...

Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/

$\text{Let}{p}_n=\text{the}{n}^{\text{th}}\text{prime}.$

$\text{Then}2025=81\cdot 25=3^4{5}^2$

$={p_2}^4{p_3}^2={p_2}^{{p_1}^{p_1}}{p_3}^{p_1}={p_{p_1}}^{{p_1}^{p_1}}{p_{p_2}}^{p_1}={p_{p_1}}^{{p_1}^{p_1}}{p_{p_{p_1}}}^{p_1}$

No information is lost by dropping the terminal 1s. Thus we may write the following form.

$$2025={p_p}^{p^p}{p_{p_p}}^p$$

The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.

Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png

Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png

Reference —

Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

Riffs and Rotes • Happy New Year 2025

No information is lost by dropping the terminal 1s.  Thus we may write the following form.

The article linked below tells how forms of that sort correspond to a family of digraphs called riffs and a family of graphs called rotes.  The riff and rote for are shown in the next two Figures.

Riff 2025

Rote 2025

Reference

• Riffs and Rotes

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

(via Vic Reiner)

An article by Ann Schilling on Diamond OA journals in combinatorics, and in particular the genesis of the journal Combinatorial Theory:

https://www.ams.org/journals/notices/202501/noti3040/noti3040.html

A complete $k$-partite graph consists of $k$ sets of vertices. There are no edges connecting vertices in the same set. Each vertex in the graph is connected to $\mathit{a}\mathit{l}\mathit{l}$ the vertices outside its own set.

In a perfect matching a subset of edges are chosen such that each vertex of the original graph belongs to exactly one of the chosen edges.

In this thread I will ask some questions about the "typical" perfect matching for $k$-partite graphs with asymptotically large numbers of vertices.