I find the presentation a bit confusing and I'm already very familiar with this material.
From a purely mathematical point of view, why this choice of topics? You correctly point out that counting partitions has no closed formula, but there are a lot of related problems (Sitrling numbers of the two kinds for example) which are of more practical utility (e.g. they are related to sums of powers formulas).
If that's too advanced for your audience then why not present more standard tricks like combinations with replacement aka stars and bars?
From a programmer's point of view, you could have focused more on how to generate subsets, permutations, partitions etc. in a memory-efficient way, for example how the Python stdlib does it.
Also, the factorial number system and the binomial base of univariate polynomials are definitely not "alternatives to base 2 in computer architectures".
Don't take this the wrong way but I struggle to see who you are writing for.
FWIW there is a way to count partitions with a fixed “basis”. Eg I wrote a post here on how to count the number of ways to make change for a given amount of money:
I am just a simple web developer and I found this stuff interesting. Learning about array programming has taught me the importance of combinatorics and its possible application to GPU programming. It is just the first chapter in a series of articles, I'd say let the man cook.
What Every Programmer Should Know About Enumerative Combinatorics -> Nothing.
It can be interesting to know something (or even a lot) about enumerative combinatorics, and certainly there are some specific programming contexts in which that's a hard prerequisite, but it's not a topic that necessarily concerns every programmer.
OTOH I think it would greatly help programmers, especially beginners, to have fewer click baity titles around.
This is easily verified by the notion that the overwhelmingly vast majority of programmers (myself included) probably know very little of the topic. Seemingly without that causing a lot of issues.
IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff for the simple reason that it's been decades since I last needed that knowledge. Very basic stuff even. Like highschool trigonometry (did some stuff with that a while back). Most programmers are just glorified plumbers that stick things together that others have built. They aren't designing new databases (for example) but simply using them. Which tends to be a lot easier. Though it helps to understand their general design and limitations. And if you are going to build a database, you might want to read up on a thing or two.
There is a wide range of esoteric topics you can dive into and learn a lot about. Diving into some of those in university is useful because it prepares you for a lifetime of needing to learn to wrap your head around random weird shit constantly that you need to understand to do the job. The point is not learning all that stuff upfront but simply learning enough that you can learn more when you need to. So, studying math and some other topics is a good preparation for that. You'll forget most of it if you don't use it. But when you need to, refreshing what you knew isn't that big of a deal.
The skill isn't in knowing that stuff but in being able to master that stuff.
Certain fields need it more than others. Graphics and vide game development needs more math than web app development (well, usually. Sometimes you need to implement a formula), including trigonometry.
I used a bunch of trigonometry when I was making 2D action games, getting characters to move about the screen and move smoothly at all sorts of angles, for one example. I also used Sine functions a lot for UI animations, making things looks like they're hovering or oscillating up and down.
I think one of the benefits of these classes, though, and university classes in general, is that even if you don't use or really remember the specifics decades later, you're at least aware of how these problems can be solved, and can look up and verify potential solutions much quicker than if you hadn't ever been exposed to it at all.
There's value in "being able to master that stuff," and there's value in "having mastered that stuff." The latter lets you trim a lot of possible designs from your search space nearly instantly, letting you focus on routes which are actually viable. The former is only of similar power when you know the design in advance or there aren't many possible solutions.
For a simple example, suppose you need to operate on `n` permutations of an enormous collection of data (far more than fits in RAM or disk), and you need those permutations to be re-usable.
One simple solution is to shuffle the indices `n` times and store the results in your cluster, but even the shuffle process is slow with normal techniques because of inter-machine random-access bandwidth issues. When using those shuffled indices for anything, you're again bandwidth-limited if the task doesn't require access of every index.
With just a tiny bit of a math background, you'll recognize that an O(1)-state shuffle is possible, where you can create some `Permutation` object with a `permute()` method, taking in an index and outputting the corresponding index in your hypothetical shuffle. That permutation will be CPU-bound rather than bandwidth-bound.
The problem with "being able to master stuff" is that your search process in the design space is slow. If I went and told you that an O(1)-state shuffle existed and would be good for the problem, sure, you'd be able to go code that up without issues. What's the chance that you'd even know to try though?
> wide range of esoteric topics ... prepares you for a lifetime of learning
That's part of it, but each of those esoteric topics also give new ideas something to latch onto. Our brains are associative, and being able to look at a new thing and tie it to a few esoteric concepts is a bit of a superpower, even if the association is weak. The difference between knowing nothing other than how to learn and knowing what's vaguely potentially possible or not is weeks or months of research. It's the difference between having to do the dumb, slow thing and being the person promoted for saving $1m/yr fixing whatever you wrote. You can get by for a long time, maybe your entire career, just making shit work, but if you're looking for more money or prestige then there are better routes.
What "O(1)-state shuffle" could you possibly be talking about? It takes `O(nlogn)` space to store a permutation of list of length n. Any smaller and some permutations will be unrepresentable. I am very aware of this because shuffling a deck of cards correctly on a computer requires at least 200 random bits.
If the requirements are softer than "n random permutations", there might be a lot of potential solutions. It is very easy to come up with "n permutations" if you have no requirements on the randomness of them. Pick the lowest `k` such that `n < k!`, permute the first k elements leaving the rest in place, and now you have n distinct permutations storeable in `O(log(n)` (still not O(1) but close).
I know this is not really your point, but misusing `O(1)` is a huge pet peeve of mine.
It's O(1) if you don't need access to every permutation (common in various monte carlo applications). 64-128 bits of entropy is good enough for a lot of applications, and that's all you get from any stdlib prng, so that's what I was comparing it to.
Those sorts of applications would tend to not work well with a solution leaving most elements in the same place or with the same relative ordering.
> IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff
The value isn't in knowing how to do math, it's in knowing when.
The value of a math class is far less in learning and remembering exactly how and, far more in learning what you can do with it so you can spot possible solutions when they arise. Expanding your mental toolkit.
This has the flavor of a post written by a programmer who got a particularly interesting interview question. They continued working on it when they got home. Hey, it's happened to the best of us (I was once asked to write "tic tac toe" and at the time I was really getting into functional programming and simple data structures, and for some reason I didn't stop working on the problem for a few days because I wanted to generalize tic-tac-toe in scale, depth, e.g. deeply nested tic-tac-toe, and dimension, where the board has more than 2 dimensions) but I'm not sure I would have written a paper "What every programmer should know about implementing nested tic-tac-toe on large arrays in high dimensions" because the answer is, almost certainly, nothing.
For those who are not aware, knowing how to count the number of structures is (nearly) the same as knowing how to encode said structure as an integer (space-optimally) (the naive general algorithm would not have good time complexity); to convert to integer, simply define some ordering on said structures, enumerate until you find your structure, the index of the structure is the integer encoding; conversely, to decode, simply list all structures and pick the structure with the index of the integer.
Just about everything that a non-specialist in combinatorics needs to know about counting can be found in Rota's twelvefold way, which lists the 12 counting problems that you can define for finite sets and shows how to solve them:
I think the best single thing for programmers to know about combinatorics is combinatorial data structures. If you have a collection of objects and you need to iterate over all possible combinations, subsets, or permutations of that collection, libraries like Python's itertools do this for you. The resulting code will be clearer, more concise, and more reliable than if you reinvent the wheel with some hand-coded rat's nest of loops.
If you need to know how long that's going to take, you'll need to know some enumerative combinatorics.
As a fan of combinatorics, I love people looking into the topic. Though most of what a programmer would find useful are approximations or trends which are handled well by the Master theorem ( https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_al... ). For those deeply interested in the deep math of combinatorics I recommend Flajolet, Sedgewick, *Analytic Combinatorics, Cambridge 2009.
I didn’t quite get from the write up what it is that every programmer should know about enumerative combinatorics or why it’s a relevant topic for programmers at all honestly.
What is the difference between integer compositions and integer partitions? It says a composition is just an ordered partition of an integer, but according to this article, number 4 can be partitioned into 5 parts, but apparently has 8 compositions. I find the 8 compositions much more accurate, but I do not get why it would not have 8 partitions.
1 + 3 and 3 + 1 are the same partition of 4, because it's a sum with the same two numbers. The order of the 1 and the 3 doesn't matter for partitions.
But that's two compositions.
I think this article could really use more applied, concrete examples if it's intended for programmers.
It seems their idea for connecting math and programming was printing out a bunch of C code, without further motivation. But coding is an applied field, developpers will want a concrete idea of what they can do with it.
I find the presentation a bit confusing and I'm already very familiar with this material.
From a purely mathematical point of view, why this choice of topics? You correctly point out that counting partitions has no closed formula, but there are a lot of related problems (Sitrling numbers of the two kinds for example) which are of more practical utility (e.g. they are related to sums of powers formulas). If that's too advanced for your audience then why not present more standard tricks like combinations with replacement aka stars and bars?
From a programmer's point of view, you could have focused more on how to generate subsets, permutations, partitions etc. in a memory-efficient way, for example how the Python stdlib does it.
Also, the factorial number system and the binomial base of univariate polynomials are definitely not "alternatives to base 2 in computer architectures".
Don't take this the wrong way but I struggle to see who you are writing for.
FWIW there is a way to count partitions with a fixed “basis”. Eg I wrote a post here on how to count the number of ways to make change for a given amount of money:
https://www.efavdb.com/change
I am just a simple web developer and I found this stuff interesting. Learning about array programming has taught me the importance of combinatorics and its possible application to GPU programming. It is just the first chapter in a series of articles, I'd say let the man cook.
What Every Programmer Should Know About Enumerative Combinatorics -> Nothing.
It can be interesting to know something (or even a lot) about enumerative combinatorics, and certainly there are some specific programming contexts in which that's a hard prerequisite, but it's not a topic that necessarily concerns every programmer.
OTOH I think it would greatly help programmers, especially beginners, to have fewer click baity titles around.
Yep, also with little in the way of motivating examples and a pile of mathematics to sift through, its hard to devote the time to this.
This is easily verified by the notion that the overwhelmingly vast majority of programmers (myself included) probably know very little of the topic. Seemingly without that causing a lot of issues.
IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff for the simple reason that it's been decades since I last needed that knowledge. Very basic stuff even. Like highschool trigonometry (did some stuff with that a while back). Most programmers are just glorified plumbers that stick things together that others have built. They aren't designing new databases (for example) but simply using them. Which tends to be a lot easier. Though it helps to understand their general design and limitations. And if you are going to build a database, you might want to read up on a thing or two.
There is a wide range of esoteric topics you can dive into and learn a lot about. Diving into some of those in university is useful because it prepares you for a lifetime of needing to learn to wrap your head around random weird shit constantly that you need to understand to do the job. The point is not learning all that stuff upfront but simply learning enough that you can learn more when you need to. So, studying math and some other topics is a good preparation for that. You'll forget most of it if you don't use it. But when you need to, refreshing what you knew isn't that big of a deal.
The skill isn't in knowing that stuff but in being able to master that stuff.
Certain fields need it more than others. Graphics and vide game development needs more math than web app development (well, usually. Sometimes you need to implement a formula), including trigonometry.
I used a bunch of trigonometry when I was making 2D action games, getting characters to move about the screen and move smoothly at all sorts of angles, for one example. I also used Sine functions a lot for UI animations, making things looks like they're hovering or oscillating up and down.
I think one of the benefits of these classes, though, and university classes in general, is that even if you don't use or really remember the specifics decades later, you're at least aware of how these problems can be solved, and can look up and verify potential solutions much quicker than if you hadn't ever been exposed to it at all.
There's value in "being able to master that stuff," and there's value in "having mastered that stuff." The latter lets you trim a lot of possible designs from your search space nearly instantly, letting you focus on routes which are actually viable. The former is only of similar power when you know the design in advance or there aren't many possible solutions.
For a simple example, suppose you need to operate on `n` permutations of an enormous collection of data (far more than fits in RAM or disk), and you need those permutations to be re-usable.
One simple solution is to shuffle the indices `n` times and store the results in your cluster, but even the shuffle process is slow with normal techniques because of inter-machine random-access bandwidth issues. When using those shuffled indices for anything, you're again bandwidth-limited if the task doesn't require access of every index.
With just a tiny bit of a math background, you'll recognize that an O(1)-state shuffle is possible, where you can create some `Permutation` object with a `permute()` method, taking in an index and outputting the corresponding index in your hypothetical shuffle. That permutation will be CPU-bound rather than bandwidth-bound.
The problem with "being able to master stuff" is that your search process in the design space is slow. If I went and told you that an O(1)-state shuffle existed and would be good for the problem, sure, you'd be able to go code that up without issues. What's the chance that you'd even know to try though?
> wide range of esoteric topics ... prepares you for a lifetime of learning
That's part of it, but each of those esoteric topics also give new ideas something to latch onto. Our brains are associative, and being able to look at a new thing and tie it to a few esoteric concepts is a bit of a superpower, even if the association is weak. The difference between knowing nothing other than how to learn and knowing what's vaguely potentially possible or not is weeks or months of research. It's the difference between having to do the dumb, slow thing and being the person promoted for saving $1m/yr fixing whatever you wrote. You can get by for a long time, maybe your entire career, just making shit work, but if you're looking for more money or prestige then there are better routes.
What "O(1)-state shuffle" could you possibly be talking about? It takes `O(nlogn)` space to store a permutation of list of length n. Any smaller and some permutations will be unrepresentable. I am very aware of this because shuffling a deck of cards correctly on a computer requires at least 200 random bits.
If the requirements are softer than "n random permutations", there might be a lot of potential solutions. It is very easy to come up with "n permutations" if you have no requirements on the randomness of them. Pick the lowest `k` such that `n < k!`, permute the first k elements leaving the rest in place, and now you have n distinct permutations storeable in `O(log(n)` (still not O(1) but close).
I know this is not really your point, but misusing `O(1)` is a huge pet peeve of mine.
It's O(1) if you don't need access to every permutation (common in various monte carlo applications). 64-128 bits of entropy is good enough for a lot of applications, and that's all you get from any stdlib prng, so that's what I was comparing it to.
Those sorts of applications would tend to not work well with a solution leaving most elements in the same place or with the same relative ordering.
> IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff
The value isn't in knowing how to do math, it's in knowing when.
The value of a math class is far less in learning and remembering exactly how and, far more in learning what you can do with it so you can spot possible solutions when they arise. Expanding your mental toolkit.
There are lots of things that we should collectively be doing as an industry but, largely, aren't.
Falsehoods Programmers Believe About Enumerative Combinatorics
Enumerative Combinatorics Considered Harmful
Enumerative Combinatorics Is All You Need
Lambda: The Ultimate Enumerative Combinator
I don't know, I once messed up a Big Tech interview question that was about enumerative combinatorics.
This has the flavor of a post written by a programmer who got a particularly interesting interview question. They continued working on it when they got home. Hey, it's happened to the best of us (I was once asked to write "tic tac toe" and at the time I was really getting into functional programming and simple data structures, and for some reason I didn't stop working on the problem for a few days because I wanted to generalize tic-tac-toe in scale, depth, e.g. deeply nested tic-tac-toe, and dimension, where the board has more than 2 dimensions) but I'm not sure I would have written a paper "What every programmer should know about implementing nested tic-tac-toe on large arrays in high dimensions" because the answer is, almost certainly, nothing.
For those who are not aware, knowing how to count the number of structures is (nearly) the same as knowing how to encode said structure as an integer (space-optimally) (the naive general algorithm would not have good time complexity); to convert to integer, simply define some ordering on said structures, enumerate until you find your structure, the index of the structure is the integer encoding; conversely, to decode, simply list all structures and pick the structure with the index of the integer.
That reminds me of https://everyuuid.com/
Just about everything that a non-specialist in combinatorics needs to know about counting can be found in Rota's twelvefold way, which lists the 12 counting problems that you can define for finite sets and shows how to solve them:
https://en.wikipedia.org/wiki/Twelvefold_way
This also takes care of most of discrete probability.
I think the best single thing for programmers to know about combinatorics is combinatorial data structures. If you have a collection of objects and you need to iterate over all possible combinations, subsets, or permutations of that collection, libraries like Python's itertools do this for you. The resulting code will be clearer, more concise, and more reliable than if you reinvent the wheel with some hand-coded rat's nest of loops.
If you need to know how long that's going to take, you'll need to know some enumerative combinatorics.
As a fan of combinatorics, I love people looking into the topic. Though most of what a programmer would find useful are approximations or trends which are handled well by the Master theorem ( https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_al... ). For those deeply interested in the deep math of combinatorics I recommend Flajolet, Sedgewick, *Analytic Combinatorics, Cambridge 2009.
I didn’t quite get from the write up what it is that every programmer should know about enumerative combinatorics or why it’s a relevant topic for programmers at all honestly.
What is the difference between integer compositions and integer partitions? It says a composition is just an ordered partition of an integer, but according to this article, number 4 can be partitioned into 5 parts, but apparently has 8 compositions. I find the 8 compositions much more accurate, but I do not get why it would not have 8 partitions.
1 + 3 and 3 + 1 are the same partition of 4, because it's a sum with the same two numbers. The order of the 1 and the 3 doesn't matter for partitions.
But that's two compositions.
I think this article could really use more applied, concrete examples if it's intended for programmers.
It seems their idea for connecting math and programming was printing out a bunch of C code, without further motivation. But coding is an applied field, developpers will want a concrete idea of what they can do with it.
This is still too abstract.
Thank you for the explanation, it definitely cleared things up for me.
> I think this article could really use more applied, concrete examples if it's intended for programmers.
> This is still too abstract.
I agree.
“Why every programmer should ignore articles with ‘every’ in the title”
What's the deal with this notice, presumably injected by GitHub into the hosted code samples in the article:
> This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below.
https://github.blog/changelog/2021-10-31-warning-about-bidir...
https://trojansource.codes/
Don't the title illustration and the actual content of the article mix up which is which?
why?
>Enumerative combinatorics is a branch of mathematics focused on counting the elements of a set.
These people are experts on sets but then they use the word "every" instead of "some" or "a subset of".
They just assume that everybody must be doing what they do. This is why Alan Kay said: “Point of View is worth 80 IQ points"
Algorithms like HyperLoglog are an exercise in Enumerative Combinatorics
To add to this HyperLogLog, sketches and bloom filters are the magic that make most of modern distributed databases tick.
that's not a helpful response. Congrats on knowing that, but the rest of us are still in the dark about the usage
Anyone else have the combinatorics song from Square One TV forever burned into their brain?
A Cyndi Lauper soundalike explains how many unique bands she can form by choosing from an expanding pool of musicians. Catchy!
https://youtu.be/w0i_ZFlGTVY?si=ThEcjtYvivMkBgcv
For me it's got to be "Nine Nine Nine".