# Questions tagged [polya-counting-theory]

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### Number of 'Unique' Ways to Mix a Color Palette?

Assume all mixtures involve colors mixed in equal amounts and base colors are already on the palette. How many ways are there of mixing k distinct colors on a paint palette where each color must be ...
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### The orbits with respect to two groups from the same conjugacy class are isomorphic

On the Wikipedia page for the Symbolic Method of Flajolet and Sedgewick https://en.m.wikipedia.org/wiki/Symbolic_method_(combinatorics) under the heading "Classes of combinatorial structures"...
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### Application of Burnside's Lemma on the vertices of a cube

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored ...
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### Number of ways to color a grid

Consider a $2000 \times 2000$ matrix which is to be filled with $15$ colors. Find the number of ways to color the matrix. Two colorings are same if rotating one of them along the axis perpendicular to ...
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### Unable to prove a result in Polya Theory in first course of combinatorices

I am trying to solve assignment problems but i am unable to prove this question. Question is -> Let c be a colouring in C, where C is set of colourings of set X on which G is a permutation group ...
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### Polya Enumeration

I have a circle divided into 60 pieces, and I have 4 different colors $(c_1,c_2,c_3,c_4)$, and I want to know how many different "colorings" I can get. So I thought I would use the Polya Enumeration ...
138 views

### Counting directed bicliques using Burnside's lemma

Let $b_{n}$ be the number of different directed $K_{n,n}$ graphs, assuming that $G$ and $H$ are considered identical when $G$ is isomorphic either with $H$ or with its transpose $H^T$ (i.e. same graph ...
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### Finding unique patter of $M$ $1$s in an $N \times N$ matrix, the rest occupied by $0$s

I am looking for a solution for a biological problem. I have a $10 \times 10$ matrix that I need to fill with $10$ molecules. They can occupy any cell in the matrix (they don't need to be next to each ...
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I need to write a Matlab code to determine the answer (which was given as 16) and I need to utilize loops to remove flips and circular shifts
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### H-graph combinatorical problem.

Consider a H-graph. It has $6$ vertices and $5$ edges (H-graph). Now let's enumerate vertices of graph with $\mathbb{Z_{+}}$ numbers , such as sum of all numbers will give us $n$. So we want to ...
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### Proper coloring of a cuboid.

Question: The different faces of a cuboid, having the measurements $1 \times 2 \times 2$ cm, is to be painted. Colors at disposal are blue, yellow, and red. In how many ways can this be done, if two ...
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### Prove that the cyclical index of this activity is expressed by the formula $I_{G_1 \bigoplus G_2}=I_{G_1}\cdot I_{G_2}$

Let $T_1$ and $T_2$ will be disjoint finite sets and let $G_1$, $G_2$ will be respectively certain permutation groups of these sets. Direct sum $G_1 \oplus G_2$ in natural way works on $T_1 \cup T_2$: ...
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### How many nonequivalent two-sided $n$-ominoes are there?

A two-sided $n$-omino is a $1$-by-$n$ board of $n$ squares with each square ($2n$ in all because of the two sides) colored with one of $p$ given colors (squares on opposite sides may be colored ...
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### Calculate in how many different ways can rotated square

We put a square $3 \text{ x } 3$ from $9$ square tiles in two types: or which can be freely rotated (we allow also symmetries). Calculate in how many different ways you can do that if we identify ...
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### Generate a list of lists that contains numbers that add up to a certain value [duplicate]

I'm not sure what kind of theorem this is called, perhaps permutation or something in the realms of that. Basically, I'm given a number and I need to generate the list of lists that contain numbers ...
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### Coloring elements of $A_1, …, A_m$ where $|A_i| = n$

Let $A_1, ..., A_m$ be separable $n$-elements sets. We color elements of these sets in use of $2$ colors, but we consider two coloring as the same if one we can get from other by changing order of ...
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### How many different colors see RG-color blinder and how many RBG-color blinder

RG-color blinder recognizes blue. He seeing objects red and green but he only knows that these colors are different (and that none of them is blue). RBG-color blinder sees red, blue and green objects ...
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### How many essentially different Drive Ya Nuts Puzzles exist?

In the Drive Ya Nuts puzzle, we are given $7$ hexagonal nuts, whose edges have been numbered from $1$ to $6$. Each nut uses all $6$ numbers. The aim of the puzzle is to place all the nuts in a ...
158 views

### removing a marble [closed]

A bag contains 3 red, 4 white, and 5 blue marbles. Jason begins removing marbles from the bag at random, one at a time. What is the maximum number of marbles he must remove to be sure that the bag ...
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### The cross entropy of Pólya Gamma distribution

Is there analytical solution for computing the cross entropy of Pólya Gamma distribution? For example, $$\int_{\omega} P_{PG}(\omega|1,c)*\log P_{PG}(\omega|1,0)d\omega$$
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### Number of $5$ full connected directed graphs

Find number of full connected directed graphs where $|V|=5$ (directed $K_5$) solution According to OEIS there are $42$ graphs like that. Let say that we have graph $G$ and we want to find how many ...
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### Counting different cube with size $2$

In set of $8$ cube (each has edge with length $1$) we have $3$ cubes with exactly one white side (other sides are black) and $5$ cubes with all white sides. We make from this cubes, one big cube with ...
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### Coloring directed wheel graph $W_6$ with $k$ colors

Let consider directed wheel graph $G_6$ example. We want to color each vertex on $1$ of $k$ colors. How many different coloring there are if two graphs we consider as the same if one can be ...
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### Coloring $n$ chain with $k$ colors

Chains are made from beads, each in one of $k$ colors. In each chain there is $n$ beads. We claim that two chains are the same if one can be made from second by cyclic rotation (mirror reflection is ...
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First of all, I am aware that these types of questions are very common and are around the internet in all shapes and sizes. However, in this case I was just really wondering if I'm doing the right ...
104 views