Questions tagged [combinatorics]

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

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1answer
46 views

Getting dataset whose cumulants match user-provided values?

I'm interested in getting list of numbers whose cumulants match user-specified list of values. Below is an example that works for list of length 2, but I'm interested in generalizing it to higher ...
3
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1answer
74 views

NextKSizePartition, or how to partition a set into subsets of size $k$ but incrementally

This question might seem very much like the linked one below but it differs in a very special way; Mr.Wizard suggested I start a new post: Partition a set into subsets of size $k$ What I want is to ...
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0answers
35 views

Symbolic Subset Sum Problem

I am interested in solving the following problem with Mathematica, but I do not know if it can be done or how it could be done. Given a finite set of terms $M=\{a_i(p_1, \ldots,p_m) \}_{i=1}^n$, ...
3
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2answers
105 views

Combinations with specific total

From the values: ...
3
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2answers
122 views

Efficient subsetting and averaging

The following code aims to average 4 groups of the values vector. Which elements belong to each group is determined by the indic ...
6
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5answers
379 views

A replacement for NextPermutation in Combinatorica [duplicate]

Does anyone know of a replacement for NextPermutation in Combinatorica? The problem with loading Combinatorica is that it interferes with new functionality which I ...
6
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2answers
131 views

Stars and Bars representation

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 $k=4$, ...
4
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4answers
212 views

Finding all the lines that can be defined a set of points

Input ten Points, calculate every possible straight line from each possible pair of points and check if any of the other points are on the lines. Is something like this possible in mathematica? and ...
22
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7answers
2k views

Permutations[Range[12]] produces an error instead of a list

This input: Permutations[Range[12]] Results in this (error) output: ...
2
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1answer
62 views

Tuples with more criteria

I have seen that there is question here which does almost what I wanted to ask but it's not quite what I wanted. Efficiently generating tuples with Outer What I would like to have is a Tuples of a ...
1
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1answer
65 views

Coloring ladder rung graphs

I can use various algorithms to list all proper $k$-colorings of the vertices of the ladder rung graph $nP_2$, the first six are show below. Is there a quick way in Mathematica to list all proper $k$-...
3
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2answers
96 views

Counting k-colorings of a graph

Combinatorica can list all $k$-colorings of the vertices of a graph $g$, which is a coloring of the vertices with no two colors adjacent, and using no more than $k$-colors: ...
6
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1answer
112 views

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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1answer
67 views

Generating summation formulas for factorized 4th moments

I'm interested in getting summation formulas for the following expression, in Einstein summation notation, with indices ranging from $1$ to $d$ $$c=X_{ik}M_{ijkl}X_{jl}$$ Here $M_{ijkl}$ is ...
7
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2answers
219 views

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 ...
4
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2answers
252 views

Scramble matrix under some condition

Assume I have a matrix. (mat = Partition[Range@9, 3]) // MatrixForm mat$=\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & ...
0
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2answers
68 views

Permutations of $ \{1, 2, \dots, n\} $ [duplicate]

When I try Permutations[Range[1, 12]]; // AbsoluteTiming I get {53.6949, Null} and with ...
4
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1answer
57 views

Counting the number of strings starting with “t” and containing 2 vowels and 2 consonants

I generated a list of strings that match the criteria in the title as follows: ...
3
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1answer
89 views

Recursive solution to the extended Josephus problem [duplicate]

The Josephus Problem is described here, with extension of killing every $k$th problem. In the simple case where every other person is killed, we can also use the binary trick. ...
2
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2answers
79 views

Finding Non-Simple Paths of a Given Length on a Graph

For the following graph: ...
4
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2answers
297 views

How to extract coefficients of a generating function like this one, using a computer?

For example if we have the generating function $G (x) = (1 + x + ... + x^k)^{10}$ and we want to calculate the coefficient of $x^{3k}$ as a function of $k $: What is the best way to go about it using ...
5
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4answers
219 views

Maximum of determinant of $n\times n$-matrix with elements 0 and 1

Generalizing https://math.stackexchange.com/questions/3265627/largest-value-of-a-third-order-determinant-whose-elements-are-0-or-1 I'd like to propose two related problems (a) find the maximum value ...
8
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2answers
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: ...
4
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1answer
115 views

Non-flat partitions of a set

A non-flat partition of a set is one where, when the elements of the set are on a grid, the partition does not contain subsets with elements from the same row. When the set is more irregular the same ...
14
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4answers
652 views

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 ...
4
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1answer
65 views

Question about FindKClique

Say we have a graph: g = CompleteGraph[5]; and that we want to find all the triangles in g. I tried to use ...
3
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1answer
62 views

Generating integer partitions with fixed first element

I'm trying to generate all integer partitions where the first entry is a fixed number i.e. all young diagrams with first row/column fixed. I'm aware of the function IntegerPartitions[n] and I could ...
0
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0answers
33 views

Triangulations of a convex polygon

I have a piece of code which finds all triangulations of a convex polygon. It is known that a number of such triangulations for a given $n$-gon is is given by ($n-2$)th Catalan Number, so for instance,...
4
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0answers
56 views

Four color theorem in Mathematica [closed]

The four color theorem is a theorem about graphs (as in graphs and networks) and it was proved with the aid of a computer. To date, there is no hand-checkable proof. The software that was used to ...
5
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2answers
224 views

Subsets of a multiset

The function Subsets[] returns the subsets of a finite set of elements. This function has a shortcoming in that it treats repeated elements distinctively. Is there ...
7
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2answers
178 views

Testing for Symmetry and Regularity in (Graph-Theoretic) Graphs

I know my way around Mathematica pretty well, however I have not been able to find any built-in functionality for testing a (graph-theoretic) graph for being symmetric (arc transitive) – this is the ...
0
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1answer
92 views

How to generate all the combinations with repetition?

I have $K$ variables. Each variable can take any value form a set with $N$ elements. We have $N^K$ possible solutions (permutations with repetition, when at each time slot we can choose among $N$ ...
1
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2answers
304 views

How to generate all the combinations? [duplicate]

There are $N$ optimization variables, $v_1,v_2,\cdots,v_N$. and $v_n\in{0,1,2,3,\cdots,K}$. Let $N=10$ and $K=5$. How can I generate all the possible combinations? For example, the first ...
0
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1answer
52 views

How make create all possible lists of twelve elements? [closed]

How can I create all possible lists of twelve elements when each element can be -1 or 1. That is, 2^12 different lists.
0
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1answer
39 views

Expanding integer compositions

Quick version: I would like Mathematica code that, for instance, turns {3,1,2} into {3,3,3,1,2,2}. More formally, given positive integers $c_1, \ldots, c_t$ which sum to $n$, produce the length $n$ ...
0
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1answer
39 views

Reverse the reccurence mirrorwise

Let be a recurrence for polynomials $R_{m,j}$ ...
0
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1answer
34 views

Different result with HypergeometricPFQ

I have a following sequence: Table[Sum[Sum[(1 + j + k + n)!/((1 + j + k) j! k! ), {j, 0, n}], {k, 0, n}], {n, 1, 5}] (* {16, 542, 31488, 2646024, 292224000} *) ...
5
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1answer
533 views

Is there a function to partition an integer set?

First I give an example. For an integer set $(0,1,2,3,4)$, there are eight kinds of subdivision or partition like this $$(0,4);\\~~(0,1)(1,4);~~(0,2)(2,4);~~(0,3)(3,4);\\ (0,1)(1,2)(2,4);~~(0,1)(1,3)(...
4
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3answers
979 views

Create all possible words using a set or letters

Given a list of letters, letters = { "A", "B", ..., "F" } is it possible to get Mathematica to generate all ‘words’ (in this example, 6 letter words), if only ...
1
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0answers
39 views

Permute using symmetric vs alternating groups

Why does this happen? Permute[{0, 0, 0}, SymmetricGroup[3]] (* {{0, 0, 0}} *) Permute[{0, 0, 0}, AlternatingGroup[3]] (* {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}} *)
2
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1answer
165 views

Rewrite the power sum in terms of convolution

Let be an identity $$n^{2m+1}=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1}k^r(n-k)^r,$$ where $A_{m,r}$ are real coefficients, see A302971 for numerators and formula of $A_{m,r}$. Here we can notice that $\...
6
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1answer
117 views

Generate list of tuples, modulo PermutationGroup

I have a permutation group, e.g. g = PermutationGroup[{Cycles[{{1, 2}}]}] but not necessarily limited to a single generating cycle. What I want is to create a ...
8
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0answers
118 views

StableMarriage vs. FindIndependendEdgeSet: How to use the procedure FindIndependendEdgeSet as a Gale-Shapley algorithm?

From Help, the procedure StableMarriage was an element of the Combinatorica, but it is available in the built-in ...
1
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3answers
127 views

Handling cases of cross terms for multi-sums

I have expressions consisting of many multi-sums and I would like to extract cross terms out of them. Consider a simple example: $$ \sum_{m_1=1}^M \sum_{m_2=1}^M \sum_{m_3=1}^M \sum_{m_4=1}^M (x_{m_1}...
3
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1answer
41 views

Using FindInstance for Identifying Feasible Planar Solutions in a 3D Latin Hypercube

Consider a $3d$ lattice latin hypercube with $n$ steps in each dimension, so it has $n^3$ positions. Coordinates $X, Y, Z \in \{1,2,...n\}$. I want to find all of the permutations of them where they ...
6
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5answers
701 views

Sum of list numbers smaller than one goal

I have five values ​​that I would like to add them so that they can be equal to or less than 3000. ...
6
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1answer
351 views

Shortest polygonal path

I'd like to find the shortest distance between some points (every point must be visited), such as between these 3 points: ...
5
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3answers
424 views

Find position without iterating

The code below solves for the four value of $m_i$ for a given pair of $(N_m,M_m)$. where $$ m_1+m_2+m_3+m_4 = M_m\\ |m_1|+|m_2|+|m_3|+|m_4| = N_m $$ Edit 2 New Sorting I have now realised that the ...
1
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0answers
37 views

Binomial coefficients for negative entries [duplicate]

Mathematica evaluates the binomial coefficient $\binom{-1}{-1}$ to 1. That agrees with an application I have in mind. However many books, such as Concrete Mathematics, e.g. see here, define $\binom{...
0
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1answer
41 views

How can I construct binomial terms using the Binomial function? [closed]

I want to construct a table of the terms Binomial[n, i] t^i (1-t)^(n-i) where i goes from 0 to ...