Questions tagged [combinatorics]

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

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7
votes
5answers
261 views

Rewriting partitions using exponents

I'm looking for a way to re-express a partition given in full form, like $\{{2, 2, 1, 1}\}$, into the shortened form $\{2^2, 1^2\}$, i.e. given a partition with repeated entries, count the number of ...
0
votes
0answers
8 views

help with proving combinatorial identity [migrated]

seems hard to prove the next identity. i think the direction is through combinatorics reasons. thanks.
9
votes
6answers
802 views

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 ...
0
votes
2answers
86 views

How to calculate and display circuits, closed paths, simple paths, cycles … of edges (with a specific length)

I'd like to know that how can I find circuit(basic), simple path, closed path of these edges: Example: ...
3
votes
2answers
150 views

Permutations with Repetition Symbol

I am trying to compute this formula in Mathematica: $$ a = \sum_{n=0}^A P_A^{A-n,n} $$ Where A can be any positive number The problem is that I am unable to find the symbol for permutations with ...
2
votes
1answer
62 views

Visualization and setting up the Kneser graph of the number of combinations “a from n by k” in Mathematica

I need to visualize the combination "a from n to k" using a graph $KG_{n,k}$ and depict the following: As we know, the graph structure is determined by the number of vertices and the connection ...
5
votes
2answers
304 views

How to solve this problem 710 of Project-Euler

I want to solve this problem: The number 6 can be written as a palindromic sum in exactly eight different ways: (1,1,1,1,1,1),(1,1,2,1,1),(1,2,2,1),(1,4,1),(2,1,1,2),(2,2,2),(3,3),(6) We ...
3
votes
1answer
37 views

Find all pairs of disjoint subsets of list

Given a set $E$, how can I find all pairs of subsets $E_1, E_2$ which are non empty and disjoint? I don't care the order of $E_1, E_2$. Right now I use a bit complicated code. First find all ...
2
votes
2answers
128 views

Counting the number of binary strings of length m with no consecutive 1s (RR). How to improve it?

I am new to Mathematica and I am trying to solve this problem of counting the number of binary strings of a certain length m, as far as no consecutive 1s are there. For instance m = 3, my recurrence ...
4
votes
3answers
151 views

All combinations assuming independence

This may be a basic question. Given an array of probabilities of length $n$, trying to create a vector of all $2^n$ combinations assuming independence between them, where either $p_i$ or $1-p_i$ is ...
1
vote
0answers
68 views
1
vote
1answer
76 views

How to speed up the calculation of the number of $4 \times 4$ Young tableaux

I find the problem of calculating $n \times n$ Young tableaux from here. I can get the number of $3\times 3$ Young tableaux by violent enumeration is $42$: ...
4
votes
3answers
186 views

Combinations to crack code

I have a set of numbers from 0 to 9 numbers=Range[0,9,1] I want to determine the combination of three numbers between 0 and 9. There is a correct combination ...
6
votes
2answers
2k views

All possible topological orderings of a graph

TopologicalSort[] returns one of many unique orderings. From wikipedia: if a topological sort does not form a Hamiltonian path, the DAG will have two or ...
4
votes
3answers
79 views

every set intersection for every set in a family with another family of sets

I want to find for each list within a list of lists what intersections occur when taking set intersection for each list in another list of lists. Hopefully that makes sense. I have tried ...
12
votes
7answers
580 views

Generate only unique combinations when input contains duplicates

I have a list with repeated elements, such as list = {a, a, b, c, c, c} and I'd like a list of the unique ways to choose 3 elements from it: ...
8
votes
2answers
230 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 ...
4
votes
1answer
60 views

Generate all spanning trees of the complete graph

How can you use Mathematica to generate all the spanning trees of the complete graph? One can count the spanning trees of a connected graph ${G}$ using e.g. the Tutte polynomial $T_{G}(1,1)$. For the ...
2
votes
1answer
45 views

Add Edge Label to certain edge in a bipartite graph

From https://mathematica.stackexchange.com/a/109436/70384 , I modified the code to display the bipartite graph I want and how do I label my edge with the edge weight by manually specifying it? I use <...
0
votes
1answer
82 views

How can I make a list of 3-connected graphs with 100 points

I'm trying to make the list of all 2-connected subcubic graphs with at most N vertices. I tried this code: ...
3
votes
1answer
53 views

Permutations and Combinations of Binary Values

I am working on optimization of multipole magnets. Reduction of the number of possible configurations of the magnets essentially becomes the unique determination of the minimum value for all possible ...
1
vote
3answers
43 views

Generate a list of the product of combinated terms in groups of 2

Here is what I have: j1=2 Do[Print[w[m1, n1] = ToExpression["w" <> ToString[m1] <> ToString[n1]]], {m1, 1, j1}, {n1, 1, j1}] That's what I got:...
2
votes
3answers
274 views

How to use Mathematica to solve this problem of planting tree

To plant trees at the center of each small square in a 3 * 4 rectangular area, it is required that there should be no continuous number of three (or more) trees in three directions of Horizontal, ...
1
vote
1answer
83 views

Manage to save large arrays

Some time ago I asked the following question: Merge list repeating elements I was easily answered and I was satisfied by the answer. However in computing such combinatorics, I saturated the RAM very ...
8
votes
1answer
137 views

Merge list repeating elements

Suppose I have a list with mixed elements {{{a,b},{c,d}},{{e,f},{h,i}}} Is there a way so that I can reshuffle the elements to get the following? ...
7
votes
2answers
752 views

All possible combinations of ways to write an equation

I want to define a function in Mathematica where I get all combinations of an expression, e.g., input: {q,p1,p2} Output: ...
0
votes
0answers
32 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 ...
1
vote
1answer
111 views

Enumerating all orientations of an undirected graph

Given an undirected graph $G$, an orientation of $G$ is a directed graph obtained by assigning every edge a direction, a superorientation of $G$ is a directed graph obtained by orienting every edge in ...
0
votes
4answers
190 views

What does GraphData[“Cubic”, n] do?

I don't have Mathematica, yet, but I just wanted to know what the output of the following was: GraphData["Cubic", 20] I'm trying to understand how many unique not-...
6
votes
1answer
141 views

how to efficiently generate all directed graphs with 6 vertices

I am attempting to generate all non-isomorphic directed graphs with 6 vertices using command ListGraphs[6, Directed].But it does not work efficiently and takes a ...
2
votes
1answer
64 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: ...
5
votes
2answers
146 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, ...
2
votes
1answer
84 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?
3
votes
4answers
98 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. ...
8
votes
3answers
378 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 ...
4
votes
1answer
66 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 ...
4
votes
3answers
324 views

Scramble matrix under some condition

Assume I have a matrix. (mat = Partition[Range@9, 3]) // MatrixForm mat$=\left( \begin{array}{ccc} \color\red1 & \color\red2 & \color\red3 \\ \color\...
3
votes
1answer
99 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 ...
0
votes
0answers
58 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
2answers
112 views

Combinations with specific total

From the values: ...
3
votes
2answers
125 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
votes
5answers
395 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
2answers
157 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
226 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 ...
23
votes
7answers
2k views

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

This input: Permutations[Range[12]] Results in this (error) output: ...
2
votes
1answer
69 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
vote
1answer
73 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
108 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
votes
1answer
125 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
71 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 ...

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