Questions tagged [combinatorics]
Questions on the use of Mathematica in combinatorics, including the Combinatorica add-on package.
357
questions
0
votes
0answers
28 views
Evaluation control in constructing table iterators
I'm trying to set up a table to scan over combinatorially many sets of numbers.
I want to have all choices of $ ({}^n C_k)\cdot({}^n C_{k}-1)\cdot\ \cdots\ \cdot ({}^n C_{k}-m) $, where I keep ...
2
votes
1answer
61 views
Delete duplicates when cycle both position and element
Cycle of position means: {a, b, a, a} and {a, a, b, a} is the same.
Cycle of element means: ...
2
votes
1answer
82 views
Building matrices of 0’s and 1’s
How would I go about computing the number of ways an nxn matrix of 1’s and 0’s can be made such that every nxm submatrix has more rows containing 1’s than the number of columns m?
5
votes
2answers
143 views
Enumerating $4 \times 4$ matrices satisfying parity constraints
I've encountered a problem, which requires computer aid, but it seems a little above my Mathematica prowess because it requires counting objects satisfying some simple conditions. It seems doable, ...
8
votes
3answers
355 views
How can I get all 4 × 4 submatrices of an n × n matrix?
I have a square matrix, I need to extract all possible combinations of 4 × 4 submatrices, where $n > 4$. For example in the case of a 6 × 6 matrix, there are 15 4 × 4 submatrices. I need the list ...
3
votes
4answers
94 views
How to set a custom number field to solve this equation
x1 , x2 , x3 , x4 and x5 can only be taken from {-1,1,2,4}.
How to set a custom number field to solve this equation.
...
4
votes
1answer
64 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 ...
0
votes
0answers
39 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
votes
1answer
77 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 ...
3
votes
2answers
108 views
6
votes
2answers
149 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
votes
4answers
216 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 ...
2
votes
1answer
63 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 ...
6
votes
5answers
381 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
votes
1answer
115 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 ...
1
vote
1answer
69 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 ...
8
votes
2answers
231 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 ...
0
votes
2answers
69 views
Permutations of $ \{1, 2, \dots, n\} $ [duplicate]
When I try
Permutations[Range[1, 12]]; // AbsoluteTiming
I get
{53.6949, Null}
and with ...
1
vote
1answer
68 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
votes
2answers
101 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:
...
4
votes
1answer
61 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
votes
1answer
96 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
votes
2answers
86 views
4
votes
2answers
299 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
votes
4answers
229 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 ...
4
votes
1answer
120 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 ...
4
votes
1answer
66 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
votes
1answer
64 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
votes
0answers
34 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
votes
0answers
59 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 ...
0
votes
1answer
130 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
vote
2answers
393 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
votes
1answer
53 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
votes
1answer
44 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
votes
1answer
39 views
0
votes
1answer
35 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
votes
1answer
535 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
$$(0,4);\\~~(0,1)(1,4);~~(0,2)(2,4);~~(0,3)(3,4);\\
(0,1)(1,2)(2,4);~~(0,1)(1,3)(...
4
votes
3answers
1k views
Create all possible words using a set or letters
Given a list of letters,
letters = { "A", "B", ..., "F" }
is it possible to get Mathematica to generate all ‘words’ (in this example, 6 letter words), if only ...
1
vote
0answers
39 views
Permute using symmetric vs alternating groups
Why does this happen?
Permute[{0, 0, 0}, SymmetricGroup[3]]
(* {{0, 0, 0}} *)
Permute[{0, 0, 0}, AlternatingGroup[3]]
(* {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}} *)
2
votes
1answer
167 views
Rewrite the power sum in terms of convolution
Let be an identity
$$n^{2m+1}=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1}k^r(n-k)^r,$$ where $A_{m,r}$ are real coefficients, see A302971 for numerators and formula of $A_{m,r}$. Here we can notice that $\...
8
votes
0answers
122 views
StableMarriage vs. FindIndependendEdgeSet: How to use the procedure FindIndependendEdgeSet as a Gale-Shapley algorithm?
From Help, the procedure StableMarriage was an element of the Combinatorica, but it is available in the built-in ...
3
votes
1answer
41 views
Using FindInstance for Identifying Feasible Planar Solutions in a 3D Latin Hypercube
Consider a $3d$ lattice latin hypercube with $n$ steps in each dimension, so it has $n^3$ positions. Coordinates $X, Y, Z \in \{1,2,...n\}$. I want to find all of the permutations of them where they ...
1
vote
3answers
134 views
Handling cases of cross terms for multi-sums
I have expressions consisting of many multi-sums and I would like to extract cross terms out of them. Consider a simple example:
$$
\sum_{m_1=1}^M \sum_{m_2=1}^M \sum_{m_3=1}^M \sum_{m_4=1}^M (x_{m_1}...
6
votes
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.
...
1
vote
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{...
5
votes
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 ...
0
votes
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 ...
1
vote
1answer
69 views
How to find efficiently the independent vertex sets from a large adjacency matrix?
I have a binary adjacency matrix $M$ of size $72\times 72$. I would like to find all possible combinations of 18 non-adjacent nodes.
There are $^{72}C_{18}$ possible (adjacent and non-adjacenct) ...
6
votes
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 ...
0
votes
1answer
48 views
Sums of binomial expressions returns different and sometimes indeterminate expressions
A problem that I encountered multiple times when taking sums over expressions involving binomials, factorials, etc. is that Mathematica (version 11.2) returns indeterminate expressions when it should ...
