Questions tagged [combinatorics]

For challenges involving combinatorics.

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Generate a Kirkman triple system

Given a universe of \$v\$ elements, a Kirkman triple system is a set of \$(v-1)/2\$ classes each having \$v/3\$ blocks each having three elements, so that every pair of elements appears in exactly ...
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18 votes
9 answers
1k views

Cryptic Multiplications

Given two non-negative integers e.g. 27, 96 their multiplication expression would be 27 x 96 = 2592. If now each digits is ...
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  • 2,153
7 votes
4 answers
260 views

Generate number set with conditions using n numbers

Generate \$T=\{T_1,...,T_x\}\$, the minimum number of \$k\$-length subsets of \$\{1,...,n\}\$ such that every \$v\$-length subset of \$\{1,...,n\}\$ is a subset of some set in \$T\$ Here, \$n > k &...
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  • 179
19 votes
23 answers
2k views

Every possible pairing

Given an positive even integer \$ n \$, output the set of "ways to pair up" the set \$ [1, n] \$. For example, with \$ n = 4 \$, we can pair up the set \$ \{1, 2, 3, 4\} \$ in these ways: \$...
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17 votes
13 answers
1k views

Enumerate all pure sets

In set theory, a set is an unordered group of unique elements. A pure set is either the empty set \$\{\}\$ or a set containing only pure sets, like \$\{\{\},\{\{\}\}\}\$. Your challenge is to write a ...
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21 votes
24 answers
3k views

Consecutive coin flips

This is a cross-post of a problem I posted to anarchy golf: http://golf.shinh.org/p.rb?tails Given two integers \$ n \$ and \$ k \$ \$ (0 \le k \le n) \$, count the number of combinations of \$ n \$ ...
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5 votes
0 answers
244 views

Sort my Cups︎︎︎︎︎︎︎︎︎︎ [closed]

I have a set of colored plastic cups. They come in four colors: green, yellow, pink, and blue. When I put them on my shelf, I like to stack them in a certain pattern. Your job is, given a list of any ...
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12 votes
14 answers
645 views

Increasing permutation trees

For this challenge a "binary tree" is a rooted tree where each node has 0 children (leaf) or 2. The children of a node are unordered, meaning that while you might draw the tree with left ...
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13 votes
12 answers
724 views

Count alternating permutations

An alternating permutation is a permutation of the first \$ n \$ integers \$ \{ 1 ... n \} \$, such that adjacent pairs of values in the permutation alternate between increasing and decreasing (or ...
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  • 22.2k
7 votes
1 answer
217 views

Find run ascending lists faster

In this question I asked you to determine if a run ascending list could be made. It was code-golf so naturally most the answers are very slow. But what if we want it to be fast. In this challenge I ...
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16 votes
16 answers
1k views

Divisible subset sums

Inspired by the recent 3Blue1Brown video Consider, for some positive integer \$n\$, the set \$\{1, 2, ..., n\}\$ and its subsets. For example, for \$n = 3\$, we have $$\emptyset, \{1\}, \{2\}, \{3\}, \...
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14 votes
3 answers
843 views

Sniff out biased random permutations

The brilliant engineers at <enter company you love to hate> have struck again. This time they've "revolutionised" the generation of random permutations. "Every great invention is ...
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25 votes
9 answers
905 views

Generate all groupings

Let's define a grouping as a flat list, which is either: just 0 2 groupings followed by the literal integer 2 3 groupings ...
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21 votes
14 answers
1k views

Verify a Superpowerset

A superpowerset (analogous to superpermutation) on \$n\$ symbols is a string over the alphabet \$\{1,2,...,n\}\$ such that every subset of \$\{1,2,...,n\}\$ appears as a substring (in some order). For ...
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  • 311
15 votes
4 answers
530 views

Output a Steiner quadruple system

A Steiner quadruple system \$SQS(n)\$ is a collection of subsets (blocks) of size 4 of a set \$S\$ of size \$n\$ such that every subset of \$S\$ of size 3 is in exactly one block. It is easy to show ...
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16 votes
13 answers
1k views

Square-free words of a length

A square-free word is a word consisting of arbitrary symbols where the pattern \$XX\$ (for an arbitrary non-empty word \$X\$) does not appear. This pattern is termed a "square". For example, ...
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31 votes
14 answers
2k views

Iterate your way to a fraction

I recently learned from a comment by MathOverflow user pregunton that it is possible to enumerate all rational numbers using iterated maps of the form \$f(x) = x+1\$ or \$\displaystyle g(x) = -\frac ...
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16 votes
13 answers
1k views

Number of complete rhyme schemes

A rhyme scheme is the pattern of rhymes at the end of the lines in a poem. They are typically represented using letters, like ABAB. We consider two rhyme schemes ...
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5 votes
0 answers
124 views

Matching fuzzies [closed]

Introduction Congratulations! You've been selected to do research a a newly discovered animal called a fuzzy, a docile, simple creature that strongly resembles a cotton ball. Fuzzies love to be near ...
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  • 3,410
18 votes
2 answers
616 views

Counting universal n-ary logic gates

Background A classical logic gate is an idealized electronic device implementing a Boolean function, i.e. one that takes a certain number of Boolean inputs and outputs a Boolean. We only consider two-...
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28 votes
25 answers
2k views

Converge to a number

Your challenge is to, given a positive integer n, count up to each digit of it, giving the effect of converging on it. Basically, count up to the first digit of n by its place value (\$⌊\log_{10}\left(...
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16 votes
11 answers
2k views

Write a number in overflowed binary

We all know how binary conversion works: the sequence of bits $$ b_1, b_2, ..., b_{n-1}, b_n $$ encodes the number $$ b_1 \times 2^{n-1} + b_2 \times 2^{n-2} + ... + b_{n-1} \times 2^1 + b_n \times 2^...
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  • 22.2k
16 votes
14 answers
956 views

AoCG2021 Day 5: Balancing sleigh with lots of trunks

Part of Advent of Code Golf 2021 event. See the linked meta post for details. The story continues from AoC2015 Day 24, Part 2. Here's why I'm posting instead of Bubbler To recap: Santa gives you the ...
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9 votes
3 answers
538 views

rank and unrank arrays of integers

Consider all arrays of \$\ell\$ non-negative integers in the range \$0,\dots,m\$. Consider all such arrays whose sum is exactly \$s\$. We can list those in lexicographic order and assign an ...
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15 votes
7 answers
3k views

Make S + S + ... + S as Large as Possible!

Let \$S \subset \mathbb N_{\geq0}\$ be a subset of the nonnegative integers, and let $$ S^{(k)} = \underbrace{S + S + \dots + S}_{k\ \textrm{times}} = \{ a_1 + a_2 + \dots + a_k : a_i \in S\}. $$ For ...
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21 votes
2 answers
775 views

Lean golf: Pascal vs. Fibonacci

The Pascal's triangle and the Fibonacci sequence have an interesting connection: Source: Math is Fun - Pascal's triangle Your job is to prove this property in Lean theorem prover (Lean 3 + mathlib). ...
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24 votes
62 answers
3k views

Implement the hyperfactorial

The objective Given the non-negative integer \$n\$, output the value of the hyperfactorial \$H(n)\$. You don't have to worry about outputs exceeding your language's integer limit. Background The ...
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  • 349
9 votes
7 answers
462 views

Distinct Subset Sums: Extending A276661

Consider the integer set \$S = \{3, 5, 6, 7\}\$. If we list all \$2^n\$ subsets of \$S\$ (its powerset) and calculate their sums, we get $$ \mathcal{P}(S) = \{\emptyset, \{3\}, \{5\}, \{6\}, \{7\}, \{...
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15 votes
6 answers
523 views

Counting maximal domino placements

Background A maximal domino placement (MDP) on a rectangular grid is a non-overlapping placement of zero or more dominoes, so that no more dominoes can be added without overlapping some existing ...
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19 votes
19 answers
5k views

What's my PIN number?

My PIN number is 1077, but that's too difficult to remember. I know from muscle memory that it's a digit, followed by a different digit, then followed by two of the same digit, which is different to ...
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19 votes
3 answers
377 views

A multiple of n in every base!

In November 2019, Alon Ran published a particularly lovely sequence in the OEIS, A329126: \$a(n)\$ is the lexicographically earliest string of digits which yields a multiple of \$n\$ when read in any ...
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14 votes
3 answers
304 views

Counting uniquely solvable polylinks

Related: Counting polystrips Background Link-a-Pix is a puzzle on a rectangular grid, where the objective is to reveal the hidden pixel art by the following rules: Connect two cells with number N ...
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  • 64.8k
23 votes
13 answers
1k views

Generalised multi-dimensional chess knight's moves

Multi-dimensional chess is an extension of normal chess that is played on an 8x8x8x8... "board". In normal 2D chess, a knight's move is a movement by a vector of \$ \begin{bmatrix} \pm 2 \\ \...
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10 votes
7 answers
402 views

Combinatorial Pipes

You're a plumber working on a house, and there's some pipes that must be connected at weird angles. You have 8°, 11.25°, 22.5°, 45°, and 90° fittings at your disposal, and you want to use as few as ...
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7 votes
1 answer
482 views

Compute the size of intersections of sets

Input A positive integer N representing the size of the problem and four positive integers v, x, y, z. Output This is what your code should compute. Consider a set of N distinct integers and consider ...
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25 votes
19 answers
1k views

Just Enough Ones

Challenge Given a positive integer \$n\$, count the number of \$n\times n\$ binary matrices (i.e. whose entries are \$0\$ or \$1\$) with exactly two \$1\$'s in each rows and two \$1\$'s in each column....
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  • 6,256
13 votes
5 answers
309 views

No More Jockeys - CodeGolf Version

This challenge is inspired by the game No More Jockeys. The input is a list of tuples of natural numbers (potentially including 0), in some appropriate input format. Starting with player 0 and ...
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  • 3,195
15 votes
2 answers
391 views

Connecting the Dots: Counting n²-gons in the n×n Grid

The recent volume of MAA's Mathematics Magazine had an article "Connecting the Dots: Maximal Polygons on a Square Grid" by Sam Chow, Ayla Gafni, and Paul Gafni about making (very convex) \$n^...
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  • 8,117
13 votes
8 answers
1k views

How often will this test fail?

Preamble There was a unit test in our codebase which was shuffling a string of length \$52\$ formed from the set of letters \$[A-Z]+[A-Z]\$ and then using the first \$20\$ characters of that shuffled ...
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22 votes
3 answers
542 views

Sticky polyhexes

Background A polyhex of size \$n\$ is a contiguous shape made from joining \$n\$ unit regular hexagons side-by-side. As an example, the following image (from Wikipedia) contains all 7 distinct ...
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  • 64.8k
8 votes
3 answers
388 views

Patterns in Permutations

This fastest-code challenge is based partly on this MSE question and exists to extend some OEIS sequences, and create others. If I extend or create sequences based on this challenge, I'll link to this ...
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  • 8,117
9 votes
2 answers
283 views

Counting polydominoes

Background A polyomino of size \$n\$ is a contiguous shape made from joining \$n\$ unit squares side by side. A domino is a size-2 polyomino. A polydomino of size \$2n\$ is defined as a polyomino of ...
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  • 64.8k
2 votes
0 answers
81 views

Find the number of parenthesis combinations [duplicate]

For example, given 3 sets of parenthesis, you have: ()()() ((())) ()(()) (())() (()()) = 5 possible combinations. Challenge Program must: • Take 1 number as an ...
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18 votes
7 answers
1k views

Matching ABACABA-type patterns

(This challenge is related to the challenge "Generate the Abacaba sequence.") Zimin words (also called "sesquipowers") are an important idea in the subject of "combinatorics ...
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40 votes
0 answers
1k views

Topologically distinct ways of dissecting a square into rectangles

I was asked by OEIS contributor Andrew Howroyd to post a Code Golf Challenge to extend OEIS sequence A049021. Would be super great to get a couple more terms for [...] A049021. Kind of thing [...] ...
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  • 8,117
7 votes
11 answers
489 views

Constrained integer partition

Challenge In this challenge, all numbers are in \$\mathbb{N}_0\$. Create a function or program that, when given a number \$N\$ and a tuple of \$k\$ numbers \$(n_i)\$ (all ≤ \$N\$), returns the number ...
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18 votes
5 answers
955 views

Count all binary relations

A binary relation on a set \$X\$ is simply a subset \$S \subseteq X \times X\$; in other words, a relation is a collection of pairs \$(x,y)\$ such that both \$x\$ and \$y\$ are in \$X\$. The number of ...
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  • 8,117
17 votes
2 answers
515 views

Count unrooted, unlabeled binary trees of n nodes

An unrooted binary tree is an unrooted tree (a graph that has single connected component and contains no cycles) where each vertex has exactly one or three neighbors. It is used in bioinformatics to ...
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  • 64.8k
20 votes
5 answers
788 views

The Caged Circles

This problem will have you analyzing circles drawn on the grid, with the gridlines drawn at integer values of \$x\$ and \$y\$. Let \$\varepsilon\$ be a very small number (think, \$\varepsilon = 0.0001\...
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  • 8,117
4 votes
2 answers
399 views

Minimal Pairing

Your program should take two lists, where each entry (a positive integer) represents the number of members of some group, as input. These lists will have the same sum but may have different lengths. ...
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