# Questions tagged [polya-counting-theory]

For questions about Pólya counting theorem. AKA Pólya enumeration theorem, Redfield–Pólya theorem, Pólya counting.

100 questions
Filter by
Sorted by
Tagged with
105 views

### Icosahedron with asymmetric coloring

I am trying to determine the number of unique solutions when placing "dots" on the sides of an icosahedron. There can be up to three dots placed symmetrically on each side. The dots are ...
44 views

### Simple way to approach bead necklaces problem for equal number of beads

I was recently given this question on a practice test, and it was intended to be completed in ~2-3 minutes, for someone with intermediate stats/math knowledge: For a positive integer $n$, you have $n$...
177 views

### Counting problem about polygon triangulations

I have the following question about triangulations (by non-intersecting diagonals, and edges) of regular polygons. What is the number of triangulations of a regular n-gon, up to all symmetry (i.e. the ...
37 views

### Redfield-Polya enumeration, but the colors don't matter

Is there a version of Redfield-Polya enumeration with the added condition that you don't care which color is which? An illustrative example is: Count edge-colorings of $K_4$ modulo the group action of ...
185 views

### Coloring the Square. Burnside's lemma.

Question from my last exam: We paint the vertices of a square with five colors: A, B, C, D, and E. Two ways of painting are considered identical if one can be transformed into the other through an ...
54 views

### Polya's enumeration theorem for edge and vertex coloring combined.

Let's say we have a tetrahedron labelled as such: We want to find the number of distinct ways to color the vertices and edges, such that 2 vertices are green, 2 vertices are red, 4 edges are black ...
1 vote
116 views

### Number of pair of edges of a dodecahedron, up to rotational symmetry.

How many ways can we choose a pair of edges of a dodecahedron, up to rotational symmetry? Taking $180$ rotation we have $15$ axis of rotations. Since there are $6$ pairs of opposite faces, we have an ...
55 views

### Why does the ratio between total solutions and unique solutions to the n-Queens Problem converge to 8?

Taking a look at the total solutions and unique solutions to the n-Queens Problem posted here (How many solutions are there to an $n$ by $n$ queens problem?) I realized that the ratio between the ...
175 views

### Can we use Burnside's lemma to count partitions of an integer?

I am currently learning group theory and find that Burnside Lemma can be used to calculate partition number, although not that efficiently. Suppose we want to distribute $n$ different balls into $n$ ...
1 vote
67 views

### Arranging triplets from an urn and partitioning them.

Say we have an urn containing 3N orbs of three different colors: red, green and blue. N of them of each color. The orbs are Arranged necessarily in sets of three, we'll call them troikas. there are 10 ...
69 views

### Identifying weak partitions

Recently a problem rose to me I'm not really sure what topic it belongs to: The origin is a "labeling application". There are n objects (say files etc., n abt. 1e6). And there are t ...
1 vote
56 views

### Bracelet enumeration

Apologies if this has been asked and answered but I can't seem to find a solution! I am looking for a way to enumerate or list all of the bracelets for a given number of beads, without repetition of ...
81 views

1 vote
126 views

### Find the number of different colorings of the faces of the cube with 2 white, 1 black, 3 red faces?

I tried using Polya theorem same in this https://nosarthur.github.io/math%20and%20physics/2017/08/10/polya-enumeration.html guide, using S4 group for cube faces. But i Have a $\frac{1}{24}12w^2*b*r^3$....
200 views

166 views

### Find a generating function for unlabeled graphs

Let's take a graph(undirected) of $m$ edges and $3$ vertices. Now I want to derive a polynomial for which the coefficient of $x^m$ denotes the number of unlabeled graphs of $m$ edges. How to derive ...
100 views

### Generating Function of P Group

Take G as group of permutations on $n$ elements. It naturally acts on the set $[n]$. Compute the cycle index polynomial of this action of partitioning the permutation into blocks of your own size. We ...
206 views

### Polya's Urn: Need Help Calculating Expected Value of Concentration of Red Balls at Stage n

An urn U contains $r_0$ red balls and $g_0$ green balls. At each stage a ball is selected at random from the urn, we observe its color, we return it to the urn and then we add another ball of the same ...
161 views

### Coloring Two Faces of an Icosahedron

Suppose that you have a solid regular icosahedron (a polytope with 20 sides all of which are equilateral triangles), and all the sides are white. (a) In how many ways can exactly two sides be painted ...
How can we calculate the total number of paths joining any pair of points in a collection of $N$ points? Specifically, consider the following example of a set of four points (their distribution on a ...