# Questions tagged [combinatorics]

Questions on the use of Mathematica in combinatorics, including the Combinatorica add-on package.

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### Looking for "Longest Common Substring" solution

I'm looking for robust code to solve the "Longest Common Substring" problem: Find the longest string (or strings) that is a substring (or are substrings) of two or more strings. I can just code it ...
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### Partition a set into subsets of size $k$

Given a set $\{a_1,a_2,\dots,a_{lk}\}$ and a positive integer $l$, how can I find all the partitions which includes subsets of size $l$ in Mathematica? For instance, given ...
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### Permutations[Range[12]] produces an error instead of a list

This input: Permutations[Range[12]] Results in this (error) output: ...
• 1,017
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### Lazy lists of Tuples and Subsets

I'm trying to build a lazy list that evaluates the n'th m-tuple or subset of a given list using Mathematicas ordering without calculating all the Tuples. The purpose is to allow for example the ...
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### How to load a package without naming conflicts?

This question applies to any package, but I encountered this problem while working with graphs. There are symbols in the Combinatorica package (such as ...
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### Finding all partitions of a set

I'm looking for straightforward way to find all the partitions of a set. IntegerPartitions seems to provide a useful start. But then things get a bit complicated. ...
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### Lazy form of Tuples/Outer to loop over list of lists

This is less a question and more asking if someone has implemented this already, with more skill. I need to perform the Outer-like generalized outer product of a ...
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### Finding all length-n words on an alphabet that have a specified number of each letter

For example, I might want to generate all length n=6 words on the alphabet {A, B, C} that have one ...
• 1,083
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### Insert $+$, $-$, $\times$, $/$, $($, $)$ into $123456789$ to make it equal to $100$

Insert $+$, $-$, $\times$, $/$, $($, $)$ into $123456789$ to make it equal to $100$. Looks like a question for pupils, right? In fact, if the available math symbols are limited to addition ($+$), ...
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### How to Derive Tuples Without Replacement

Given a couple of lists like a={1,2,3,4,6} and b={2,3,4,6,9} I can use the built-in Mathematica symbol ...
• 1,029
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### Find all permutations with a condition

How can I find all the permutations of {a, b, c} where a + b + c = n? For instance: if n = 3...
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### Faster derangements?

I wonder what is the fastest method to generate derangements? Both the Combinatorica function and Martin Ender's answer to Permutations of lists of fixed even numbers are based on filtering the ...
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### Shuffle product of two lists

I want to do the following: I have two lists {a_1,...a_n}, {b_1,...,b_n} and I would like to build now all shuffles out of this. This means all unions of these ...
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### How to count all cliques (not just maximal ones) in graphs?

How can I count the numbers of $r$-cliques in a graph? In other words, the number of triangles, the number of $K_4$, the number of $K_5$ etc. I have tried for example ...
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### Alternative to Subsets to generate k-combinations

I need to generare a huge number of k-combinations for a quite big set of numbers (more than 100), so I would avoid Subsets because it requires too much memory (it generates all subsets at once). I ...
• 2,037
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### Partition a set into $k$ non-empty subsets

The Stirling number of the second kind is the number of ways to partition a set of $n$ objects into $k$ non-empty subsets. In Mathematica, this is implemented as ...
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### Equivalent Nested Loop Structure (combinations_with_replacement)

Consider the following examples: ...
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### Mathematica implementation of Zeilberger's algorithm (previously done in Maple)

I have this Mathematica code: ...
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### Non-descending Tuples

I want to get all the non-descending tuples of a list with given length, for example: f[{1,2,3,4},{3}] ...
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### How do I generate the upper triangular indices from a list?

I have some list {1,2,3} How do I generate nested pairs such that I get {{1,2},{1,3},{2,3}} That is, I'd like a way to ...
• 2,263
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### How to generate all Feynman diagrams with Mathematica?

I'm using the words "Feynman diagrams" for indexing purposes, but the question is in fact purely graph-theoretic. Given a list n={n1,n2,...} of non-negative ...
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### Finding elementary cycles of (directed) graphs

I need to enumerate all the simple cycles (i.e. elementary cycles where no vertex is repeated other than the starting) of a graph which has both directed and undirected edges, where we can treat the ...
• 549
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### How to correct my code for solving the Josephus problem?

Problem Description Recently, I have been reading the book Schaum's Outline of Mathematica (2nd Edition), where I encountered the problem: Flavius Josephus was a Jewish historian of the first ...
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### Is it possible to generate a Hasse diagram for a defined relation?

I'm looking for a way to create a Hasse Diagram from a given partial order binary relation. The relation will be given explicitly, for example: ...
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### Finding all sublists or substrings of a given list/string

Given a list or string, how do I get a list of all (contiguous) sublists/substrings? The order is not important. Example for lists: ...
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### Exploring all combinations of parameters

I often need to explore a large parameter space, e.g. making dozens of plots using a range of parameters. This looks something like: ...
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### Efficient lazy weak compositions

In Mathematica all weak k-compositions of integer n can be generated using permutations of integer partitions: ...
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### Solving word search puzzles

I am trying to create a code that can identify the following terms in a grid of letters: MATHEMATICA, STACK, EXCHANGE and USERS. ...
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### Improving speed of code computing number of nonrepeating partitions

I need to answer the following for a number of parameters: How many ways can the integer $k$ be written as a sum of $n$ different integers ranging from $1$ to $m$? My initial attempt was the ...
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### Generating set partition diagrams

I recently came across a very nice illustration of set partitions on wikipedia (Partition of a set article) I need to reproduce this diagram in order to modify some things, does anyone have a good ...
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### Memory efficient generation and selection of tuples

The short of this question: I need to find a bunch of n-tuples of {0,1} which do not fail to satisfy a set of (non-linear polynomial) equations, without hogging up all of my memory by trying to ...
• 243
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### Generating an efficient way to compute einsum?

Given an einsum like below, how could I generate an efficient computation graph for it? $$X_{ik} M_{ij}M_{kl} X_{jl}$$ The indices range from $1$ to $d$ and the goal is to minimize computation time ...
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### Challenge: Creating Compilable Permutations Function

As Mathematica's implemented Permutations function is not compilable, I tried to write my very own Permutations implementation, ...
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### How to find all graph isomorphisms in FindGraphIsomorphism

I found the second definition of the function FindGraphIsomorphism not working. Here's the definition Mathematica 8 gives: ...
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### How to improve the performance of solutions to Project Euler (#39)?

This is the problem 39 of Project Euler, which I asked in the chat room two days ago. My original code runs as slowly as snails, and finally I got two answers from JM and Rojo. Unfortunately, both of ...
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### How to generate all possible orderless partitions of a list according to another list?

This question is hard to describe in plain text. So I will post an example and a working code (brute force) to illustrate. For example I have a list: ...
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### Generating Tuples with restrictions

I am interested in a certain subset of all tuples. Generating all with Tuples and then filtering is wasteful, and will "blow up" very quickly. For a concrete ...
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### Partitioning with constraints on subsets

Given the following data: constraints = {{11, 2}, {11, 3}, {11, 4}, {11, 6}, {11, 9}, {1, 6}, {5, 6}, {2, 5}}; weights = {3, 7, 3, 2, 4, 2, 2, 2, 3, 2, 1}; I ...
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### How to apply a permutation to a symmetric square matrix?

Given a symmetric square matrix, how can I apply a permutation to the rows and columns (i.e. the same permutation to both the rows and the columns) such a way that the new structure of the matrix ...
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### What is the fastest way to get the nth distinct permutation of a list?

What is the fastest way to write a function nthPermutation[xs_List, n_Integer] which is equivalent to Permutations[xs][[n]] but ...
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### Determining all possible traversals of a tree

I have a list: B={423, {{53, {39, 65, 423}}, {66, {67, 81, 423}}, {424, {25, 40, 423}}}}; This list can be visualized as a tree using ...
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### Better way to get Fisher Exact?

I want to perform a Pearson's $\chi^2$ test to analyse contingency tables; but because I have small numbers, it is recommended to perform instead what is called a Fisher's Exact Test. This requires ...
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### Binomial[-1,-1]

According to various sources e.g. http://www.proofwiki.org/wiki/Definition:Binomial_Coefficient and Wolfram themselves http://functions.wolfram.com/GammaBetaErf/Binomial/02/ , the binomial ...
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### Combination and Permutation

How could I obtain the list of all the groups of 5 numbers taken from Range[12] such that the 2 lists have an empty intersection : ...
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I had asked this question before but I guess I did not make the question clear enough and I apologize for that. Here is the problem: Tuples gives me more data than ...
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### Sierpinski carpet with GraphData

Is this graph in the list among the so-called "standard" structures used in GraphData? However, I have not found yet anything like "Carpet" or "Sponge" in the list ...
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### Enumerating all connected graphs with 9 vertices

The version of my Mathematica is 10. I generate a connected graph with 9 vertices, but find that it is not included in Mathematica. ...
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How can I visualise/represent "Stars and Bars" in Mathematica? Say I have $n$ balls and $k$ slots to fill (or not to fill) with balls, e.g. when $n=4$ and $k=4$, ...