# Questions tagged [combinatorics]

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

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### Create list with integer partitions satisfying some conditions

I want to create a function PartSet[N_,M_] of two positive integer variables which outputs a list of all pairs of integer partitions for all integers up to $M$ ...
• 1,287
1 vote
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### Sorted Tuples without Filtering

Say I have a list $L$ where the elements can be sorted into some canonical order. I want to use Tuples[L,m] but I only want the output lists to be sorted and ...
• 145
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### Using the generalised binomial theorem to expand an expression

I would like to use Mathematica to compute the following expansion: $(1+x)^\rho= 1 + \rho x +\dots$ for some $|\rho|<1$ as for example explained here. I tried the Series expansion functions ...
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### Picking integer compositions with certain descent patterns

I am trying to find a nice way to pick out all the integer compositions (ordered partitions) of an integer $n$ that satisfy a given pattern of descents between some adjacent elements. Writing a ...
419 views

### Tuples optimization challenge

Consider a function that for a given integer $0\le n <256$ computes the number of leading zeros in its binary representation. ...
• 15.7k
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### Combine each element with all the others in sublists

Suppose that I have a list of numbers list = {{{1, 0}, {-1, 1}, {0, -1}}, {{-1, 0, 1}, {-1, 1, -1}, {-1, -1, 0}, {1, 1, 0}, {1, -1, 1}, {1, 0, -1}}} I would like ...
95 views

### Deleting sublists based on a criterion

I generated a list as follows ...
1 vote
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### Enumerating labeled graphs on n vertices

I'm trying to enumerate the labeled graphs on $n$ vertices having at most $e$ edges. I thought GraphData /@ GraphData[n] and then filtering by edge count would do ...
• 11
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### How can I convert sequences to sharings and vice versa?

Given positive integers $k,n$, a $k$-sequence of $I_n$ is a list of $k$ not necessarily distinct elements of $\{1,\dots, n\}$. And an $n$-sharing of $I_k$ is a list of $n$ possibly empty, disjoint ...
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### Implementing summation under combinatorial restriction

For $m,n\in\mathbb N$, I am interested in the numerical evaluation of $$f(m,n) = \sum_{s_j\in\{\pm1\}}' \prod_{k=1}^{2n-1} (1-e^{\frac{2i\pi}{m} s_k(s_{k+1}+s_{k+2}+\cdots+s_{2n})}),$$ where the ...
• 909
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### Is there a Mathematica function that generates all ordered partitions?

My book defines a length $k$ ordered partition of $I_n$ as a sequence of $k$ disjoint, possibly empty subsets of $\{1,\dots, n\}$ that union up to $\{1,\dots, n\}$. Is there a mathematica function ...
• 317
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### Mathematica code for q-Stirling numbers

In the paper [A new $q$-Analog of Stirling Numbers],(https://hal.archives-ouvertes.fr/hal-01372920/document)[PDF] J. Cigler defined $q$-Stirling numbers of the second kind as the following: He ...
• 121
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### A code that returns the partial permutations on {1,2,...,n}

A partial permutation is a bijective partial function. A partial function is a map from a subset of {1,2,...,n} into {1,2,...,n}. I want a list of the matrix representations of all the partial ...
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### Building and Plotting a discrete CDF

I'm trying to plot the CDF of a simple "nCr" experiment: A box contains 4 screws and 6 nails. Two items are drawn at random without replacement. Let X be the number of nails drawn. I built a ...
• 101
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### Producing a random Wang Tile tiling image more efficiently

I'm following along with this SIGGRAPH 2006 paper Recursive Wang Tiles for Real-Time Blue Noise - there's a video here too. Eventually I want to try to produce the blue noise results in the paper, and ...
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### Permutations with Repetition

I am working with a function of type F[a,b,c,d,e,f] that obeys the following symmetries: ...
58 views

### Conditional Statement and Loop in Mathematica to find bound

I have a question regarding loop and conditional statements. I have an equation where I would like to find the bound for n and m based on the value of h. Here is what I have so far; ...
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1 vote
59 views

### Triangulation of Point Configuration

I was going through the documentation of Mathematica but couldn't find any built-in function that can find all possible triangulations for a given set of points. For example, if I have the following ...
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137 views

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### Matroids in Mathematica?

I can't seem to find a "standard" way to implement/manipulate matroids in Mathematica. (They do not seem to be included in Combinatorica, for instance, and googling has turned up nothing.) ...
• 121
139 views

### Find different combinations of 3 lists with given constraints

My inputs are 3 lists of unequal length: A={a,b,c,d} B={i,j,k} C={v,w,x,y,z} And I want to find the combine set X which looks like ...
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### Enumerate all possible subsets such that all are of the same length which is maximum and each contains non-repeating elements?

Given a list lists: ...
• 1,556
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### Extract only a few coefficients of the multiple of extremely many polynomials

I want to extract some informations, say the coefficients of $x^{500}$ and $x^{1000}$ terms (numerical values are acceptable), of the multiple of ...
• 1,161
330 views

### Partition a nested list such that no repeated elements in every subsets?

I have a large list and for simplicity, let's take the simple list as an example: lists = {{1, 2}, {1, 6}, {2, 3}, {2, 5}, {3, 4}, {3, 6}, {4, 5}, {5, 6}} I would ...
• 1,556
1 vote
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### How do I generate a random permutation of a (long [millions-length]) list of symbols? [closed]

I have a list of length twelve: p = {t[1, 1], t[1, 2], t[1, 3], t[1, 4], t[2, 2], t[2, 3], t[2, 4], t[1, 5], t[3, 3], t[3, 4], t[1, 6], t[4, 4]} and a set of ...
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1 vote
56 views

### Construct all possible 3-letter words from A,B. Repetition of letters is allowed [closed]

I have two letters A and B. I need to construct all possible 3-letter words. Repetition of letters is allowed. I know that the answer to this problem is 2^3=8. But ...
• 229
131 views

### Fast enumeration of all perfect matchings in complete graph

I have a code that generates the list with all possible Perfect Matchings (PM) of a fully connected graph. Each edge of the graph is mono or bi-colored with up to ...
• 327
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### Finding all Latin Squares of order 5

A Latin Square is a square of size n × n containing numbers 1 to n inclusive. Each number occurs once in each row and column. An example of a 3 × 3 Latin Square is:  \left( \begin{array}{ccc} 1 &...
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1 vote
55 views

### Edge thickness in directed path graph doesn't respond

I have the following code to draw a lattice path in 3D: ...
• 1,505
429 views

### Count number of balls in each bin, given a two-element sequence of balls and bins

If I have a list: {ball,ball,BINDIVIDER,ball,ball,ball,BINDIVIDER,BINDIVIDER,ball,BINDIVIDER,ball} The balls and bins can be in any permutation. Then, the ...
• 1,505
84 views

### How to create all possible permutations? [closed]

there is a problem: I have 5 letters - a,b,c,d,e and 20 places. I have to use each letter at least once but no more than 10 times. So the result can look like {a,a,a,a,a,a,a,a,a,a,b,b,b,b,b,b,b,c,d,e},...
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1 vote
117 views

### How to generate all the combinations with repetition and another conditions? [duplicate]

I want to generate all the combinations with repetition for k variables with values from a set of n elements. There are some ways, I like this formula, which I found on this forum (it is for n = 2 and ...
• 11
1k views

### Transform a number to a factorial

I came across the need to transform a number into a factorial n, with positive integer n. I have searched in the MMA information but I can't find anything like that. I imagine an input, which verifies ...
• 2,145
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### Splitting balls over sized bins

This is strongly related to Splitting a set of integers over a set of bins, but a much simpler case. If we have $N$ indistinguishable balls and $k$ arbitrarily large distinguishable bins, we know the ...
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### Splitting a set of integers over a set of bins

I have a problem that feels like it should be simple but I'm just drawing a blank on. I've got a set of integers, e.g. ...
• 45.4k
83 views

### Generating representatives of rotation classes of tuples of ones and zeros with a fixed number of ones

In a related question I asked how to generate all the tuples of ones and zeroes with a fixed number of ones (generating tuples of ones and zeroes with a fixed number of ones). I wish to consider a ...
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446 views

### generating tuples of ones and zeroes with a fixed number of ones

I would like to generate all the tuples of ones and zeros of a given length and with a given number of ones without generating all the possible tuples, which is impossible for tuples of large enough ...
• 567
1 vote
133 views

### How to efficiently replace the repetitive sequence?

The problem is how to determine the repetitive sequences and replace the part with consecutive sequences For example: A={{1,3,4},{2,3,5},{1,6}} Then, detect there ...
• 13
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### How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?

I know just the basic operations on graphs using Mathematica. But I want to know how to write a code that prints all the possible combinations of a graph with a specified number of edges. Take for ...
580 views

### Scan through (partial) tuples

I have a list of list of positive integers $s = \{s_1, s_2, ..., s_k\}$, each list $s_i$ is possibly of different lengths, and I want to find out if there exists a $k$-tuple of the $s_i$ that sums ...
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1 vote
40 views

### Custom Table, for iterating over permutations

It is common that I iterate over all permutations of 1,2...,n, either by making a table or performing a sum. Instead of creating the set of all permutations, it would be better to iterate over them. ...
• 2,292
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### How to delete duplicate graphics of the same kind?

A054247: Number of n X n binary matrices under action of dihedral group of the square D_4. ...
1 vote
87 views

### How to solve this problem by the way of saving memory?

Eight different boys and five different girls are in a row. Girls are required to be next to each other. How many ways are there (the answer is $9!\times5!$)? ...
107 views

### What is the easiest and efficient way to get positive or negative combinations of a list?

Consider the following simple case: list={1,2,2}; Subsets[{Splice[list],Splice[-list]},{1,Length@list}] This produces the following combinations: {{1}, {2}, {2}, {-...
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### How to visually display the Stirling permutations of $k^{th}$ order?

Definition of Stirling permutation from Wikipedia: In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset $\{1, 1, 2, 2, ..., k, k\}$ (with two copies of ...
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### How to correctly enumerate all the schemes of this cube coloring problem?

This problem is the fifth question of 1996 Chinese High School Mathematics League or Chinese Mathematical Olympiad in Senior: Choose several colors from the given six different colors to dye six faces ...
591 views

### How to correctly calculate the number of seating plans for the 4-couples problem?

Four couples a are sitting around a round table, in which husband and wife can not be adjacent. How many different seating plans are there? I want to get as many simple calculations as possible. ...
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### Combinatorial selection with constraints

Five of the 10 actors can only sing, two can only dance, and three can both sing and dance. Now, how many kinds of selection methods are there to perform a program that requires two people to dance ...