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# Questions tagged [combinatorics]

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

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128 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 $k=4$, ...
209 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 ...
57 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 ...
376 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 ...
107 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 ...
66 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 ...
217 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 ...
67 views

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

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

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 $k$-...
92 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: ...
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: ...
87 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. ...
77 views

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

For the following graph: ...
294 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 ...
213 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 ...
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 ...
64 views

Say we have a graph: g = CompleteGraph; and that we want to find all the triangles in g. I tried to use ...
60 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 ...
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,...
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 ...
76 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 $N$ ...
259 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 ...
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.
36 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 $n$ ...
39 views

### Reverse the reccurence mirrorwise

Let be a recurrence for polynomials $R_{m,j}$ ...
33 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} *) ...
530 views

699 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. ...
37 views