# Questions tagged [combinatorics]

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

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### How to determine all cases consistent with constraints

Suppose that $(i, j), (k, l)$ and $(m, n)$ are pairs of non-negative integers, satisfying the following constraints: $$i < j,\;\; k < l, \;\; m < n$$ and $$(i, j) < (k, l) < (m, n)$$...
391 views

### How can I prevent these warnings while using ParallelTable

I have this code to find all the permutations of a set of letters that form legal words. ...
776 views

### Counting the number of a specific type of permutation

In the theory of cumulants of vector-valued random variables, the following types of formulas appear: \begin{equation} \theta^i \theta^{jk}  = \theta^i \theta^{jk} + \theta^j \theta^{ik} + \theta^k ...
593 views

### Solving variant of the knapsack/money-changing problem

I'm trying to solve a variant of the knapsack/changing money problem where I have a set of a few numbers and I'm trying to find the linear (integer) combinations of them which are close to a given ...
716 views

### 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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### What function returns all possible permutations with repeating list elements?

Let's say I want to find a sample space of such experiment: There are three exits from the box: Left (L), right (R) and front (F). Three mice have been put in that box. Find the sample space of an ...
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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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### How to evaluate the sum over a hyperplane

I have difficulties in evaluating the following expression: $$\sum_{\small n_1+...+n_{k}=m-k}\; \prod_{i=1}^{k}\frac{1}{(n_i+1)(n_i+2)}$$ I have tried the function ...
238 views

### Finding integer partitions one at a time

I am writing a small program and I need to calculate integer partitions of a handful of numbers. In my code I just run IntegerPartitions[k,{n}] and then iterate through the results. Since I only use ...
368 views

### Need Help Writing (a Pascal) Matrix in Mathematica

I want to write a function $f[n]$ in Mathematica which gives me an $n\times n$ lower triangular Pascal matrix with a row of zeros in between each nonzero row. That is, I want the matrices \begin{...
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### Exact cover solution

Is it possible to get a exact cover solution(s) and/or number of possible solutions in Mathematica?
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### Insert $+$, $-$, $\times$, $/$, $($, $)$ into $123456789$ to make it equal to $100$

Looks like a question for pupils, right? In fact if the available math symbol is limited to $+$, $-$, $\times$, $/$ then it's easy to solve: ...
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### (Efficiently) Generating graphs with vertex degree 3 for all vertices

I'm working on a graph theory problem, and given $n$ vertices, I would like to be able to generate all non-isomorphic connected graphs (not necessarily simple) with $n$ vertices, each having degree 3. ...
934 views

### Find all permutations with reversals / cyclic permutations removed

I have a list of all non-cyclic permutations of n labels. How can I get rid of all elements which are redundant in the sense that they are the inverse of another one. For instance if n=4, the ...
750 views

### permutation as product of transpositions

How can the permutation that takes{-f, -i, i, -e} into {-e, -i, i, -f} be realised as a sequence of nearest neighbour ...
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### Help with Permutations

I am currently trying to generate a list of permutations of length 2 of elements of a list of strings. For example: ...
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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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### Combinatorica: Girth[] and FindCycle[] disagreement

Warning: run the following code in a fresh Mma session, as some symbols could be shadowed (depending on your Mma version) While trying to answer this question, I fell into the following: ...
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### What is the fastest way to count square-free words?

Background A word is a string of letters in an alphabet. A square-free word has no adjacent repeating substring. For example, (in the ternary alphabet of {0,1,2}) the words 00, 012121, and 0212012021 ...
982 views

### 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 ...
3k views

### 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 ...
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### Word Squares and Beyond

A word square is a set of words which, when placed in a grid, read the same horizontally and vertically. For example, the following is an English word square of order 5: ...
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### Generating Linear Extensions of a Partial Order

Given a set $S$ and a partial order $\prec$ over $S$, I'm looking for a way to "efficiently" generate a list of linear extensions of $\prec$. Suppose the partial order is given by a ...
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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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### BipartiteMatching hangs with floating-point weights in Mathematica 7 (package Combinatorica)

I am trying to use Combinatorica's BipartiteMatching algorithm on a weighted graph. I need to use real numbers as weights. If I try the following ...
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### How to create tournament bracket

I'd like to create a tournament bracket using Mathematica. I've looked around online, but haven't found any examples yet. Can someone show me how to do this? Specifically, I'd like to have a (somewhat)...
5k views

### 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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### 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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### 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 ...
5k views

### 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: ...
534 views

### When to use built-in Graph/GraphPlot vs. Combinatorica

What are the pros and cons of using built-in Graph/GraphPlot (and related) types vs. types in the Combinatorica package?
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### Combinatorica Graph from Edge List

Can someone provide an example of a Combinatorica-based graph which uses ShowGraph and Graph and takes in an explicitly defined list of edges (not some auto-generated graph). I have not been able to ...
647 views

### Mathematica function/package for shuffle permutations

Does anyone knows a way to compute a list of all (p,q)-shuffles in mathematica? For a definition of the shuffle permutations see for example http://ncatlab.org/nlab/show/shuffle I'm dreaming 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 ...
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How can I calculate only those permutations of Range[n], that satisfy certain rules? I don't want to filter the result after calculating all the permutations but ...
255 views

### NumberOfSpanningTrees command not working correctly

I am attempting to use the Combinatorica NumberOfSpanningTrees command for all n-cycle graphs from 3 to 30. I am trying to get a ...
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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 ...
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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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### Efficiently Visualising Very Large Data Sets (without running out of memory)

I have put a few really hard problems in combinatorics up against Mathematica 8. I'd have to say that it works really well, until you want to view the data. If you look at my question Advanced ...
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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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### Count number of sublists with a total not greater than a given max

Suppose I have a list of positive integers: data={1, 1, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 8, 9, 10, 10, 12, 16, 23} I want to count the number of subsets up to ...
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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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### 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 ...