Questions tagged [necklace-and-bracelets]

In combinatorics, a *necklace* of length $n$ is an equivalence class of strings of length $n$, under rotation, so $abcde = bcdea$. A *bracelet* is an equivalence class of strings under rotation and reflection (so in addition $abcde = edcba$).

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Necklace combinatorics

Consider a circular necklace with $18$ identical beads. We can rotate the necklace and turn it over. Let $G$ denote the symmetry group. How many rotations does $G$ contain? How many reflections does $...
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How many necklaces can be formed with $6$ identical diamonds and $3$ identical pearls

Find number of ways to make a necklace (or a garland) consisting of $6$ identical diamonds and $3$ identical pearls. I got the correct answer $7$ by taking different cases but when I applied the ...
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How many necklaces made of black and white beads (k total, x black) have at least y consecutive black beads?

Consider all necklaces consisting of black and white beads, of length k, containing x black beads. How many such necklaces contain y consecutive black beads somewhere in the necklace? (y is less than ...
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3 answers
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Quantifying the evenness-of-distribution of nodes within a necklace

Given a necklace with n nodes that are distributed around a circle by a set of given deltas: How would you quantify how evenly the nodes are distributed. (By "evenly" I mean that each node ...
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3 votes
1 answer
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How many necklaces with given certain colored beads

I am interested to know if there's a way to calculate the number of (rotation agnostic) necklaces that can be produced from different colored beads, each color with its own quantity. For instance, if ...
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How many necklaces are there with a known number of beads of each color? [duplicate]

I suspect that the answer to my question might be trivially found in the Wikipedia page for the combinatorial concept of a necklace, but I'm finding that page very hard to understand. Suppose I have $...
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Number of different seven-bead necklaces with 4 colored beads [closed]

How many different seven-bead necklaces are possible, assuming each bead is one of four different colors and each necklace contains exactly one bead of one color and exactly two beads of the three ...
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How many different circular necklaces containing ten beads can be made using beads of at most two colors?

How many different circular necklaces containing ten beads can be made using beads of at most two colors? I know I need calculate situation with 2 colors, 8 second color, 3 first color, 7 second ...
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Find the number of different types of circular necklaces that could be made from the sets of beads

Find the number of different types of circular necklaces that could be made from the sets of beads 7 black and 5 white beads We need to solve this by the Polya-Burnside method of enumeration: Since ...
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2 votes
2 answers
282 views

Number of ways to arrange objects in a circle, some of which may be identical

I know that the number of ways to arrange $n$ distinct objects in a circle in $(n-1)!$ from Circular Permutation. But suppose we have $n_1$ identical objects of Type $1$, $n_2$ identical objects of ...
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Using burnside's lemma to calculate a smaller subset of unique, color-agnostic bracelets

We have a child's toy, which is a ball made of 12 colored wedges (3 Red, 3 Green, 3 Blue, 3 Yellow). Our child asked the sensible question 'how many different patterns are possible?'. In researching ...
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Finding "beautiful" necklaces with regular gaps

I am looking for "beautiful" arrangements of $k$-ary necklaces of length $ak$ where each of the $k$ types of bead appears $a$ times ($a \geq 1$ a natural number). A necklace is considered ...
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Interpretation of combinatorial identity involving binomial coefficients

I proved that for some positive integers $n$, $k\leq n-1$ and $l\leq k$ the following identity is satisfied. $$\binom{k-1}{l-1}\binom{n-k-1}{l-1}\dfrac{n}{l} = \binom{k}{l}\binom{n-k-1}{l-1} +\binom{k-...
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Combinatorial Necklaces & Strips of $n$ Beads and $k$ Colours

Say I have $n$ indistinguishable beads and $k$ different colours. Suppose here and for the rest of the writeup that $k \mid n$ unless otherwise stated. I want to colour all the $n$ beads using exactly ...
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Number of necklaces given 3 beads that can take 4 colours

Find thee number of necklaces given 3 beads that can take 4 colours, two necklaces are considered the same if the colours can be matched by rotation or flipping them. My approach is by using Burnside'...
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1 answer
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Burnside's Lemma necklace with a clasp

How many necklaces with clasps can be made with 6 beads in one colour, 4 in second and 3 in the third? Use Burnside's Lemma I've got: Let $X=\{\text{necklaces of length 13 beads = 6 white, 4 black and ...
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Burnside's Lemma bracelet with beads in 2 colours (uneven cardinality)

How many bracelets can be made with 5 beads in one colour and nine in the other (rotated and flipped bracelets are considered the same). I'm trying to use Burnside's Lemma yet I'm little confused. So ...
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Beads on a necklace but they’re all the same colour

Okay, so I know that if 15 beads, 5 red, 5 yellow, 5 blue, the number of possible combinations is: 14! / 2*5!*5!*5! But say all 15 were of the same colour. (the answer is obviously 1). But wouldn’t ...
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How many distinctively different necklaces are possible if you use 0 to 3 beads out of 5 different beads given? [closed]

Today in my IB Math II HL/AP Calculus BC class we were going over IB Math I topics and did the given problem. So you would calculate the number of necklaces for a 0, 1, 2, and 3 bead necklace. I used $...
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3 votes
2 answers
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Necklace problem with Burnside Lemma

How many necklaces can be made with two red beads, two green beads, and four violet beads?(8 total) Using Burnside lemma is complicated for me due to my lack of understanding of the lemma. I want to ...
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1 vote
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Circular and Reflectional Symmetry

If you have two types of objects, $a$, and $b$. You have $n_1$ of the first object, $n_2$ of the second object. How many distinct ways can you rearrange these objects on a ring. Arrangements are only ...
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A necklace is made up of 3 beads of one color and 6 beads of another color

A necklace is made up of 3 beads of one color and 6 beads of another color beads of same color are identical the number of necklace that are possible ? I have attempted the question in this way : ...
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$n-$circular arrangement problem

Find the number of ways to arrange $n$ people in a circle so that $3$ people are separated. My approach: The number of ways to arrange $n$ people in a circle is $(n - 1)!$. If the $3$ people are ...
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"bracelet type" Combinatorics

This question seems ok but I'm having real difficultly working out the answer using the method they provided. It's so hard to keep track of all the options. Does anyone know of a better more algebraic ...
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Circular permutations of identical objects of two kinds.

A necklace is made up of $3$ beads of one sort and $6n$ of another, those of each sort being similar. Show that the number of arrangement of the beads is $3n^2+3n +1$. My attempt: There are total $...
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Burnside Lemma Necklace

I'm trying to solve this problem for 4-6-8 necklace. So far I am at the following: $\frac{18!}{4!6!8!}$ (different arrangements) + $\frac{9!}{2!3!4!}$ (rotating through 180 degrees). Now I consider ...
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1 vote
1 answer
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How many different circular bracelets can be formed from 8 red, 6 blue and 4 yellow beads, always using all available beads?

How many different bracelets can be made from 8 red, 6 blue and 4 yellow beads, always using all available beads? Two bracelets are different when we cannot get the color scheme of the beads of one ...
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4 votes
1 answer
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Necklace combinations with three group of beads

I have a hard question about a way how many different necklaces can be made. Suppose that we have the following restrictions: We have 3 groups of beads: 4 triangle beads 6 square beads 8 circle ...
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Correspondence between $k$-ary Lyndon words and $(k-1)$-ary Lyndon words without repetitions

At Counting Lyndon words with no adjacent character repeats, it turned out that for $n\ge3$ the number of $k$-ary Lyndon words of length $n$ without adjacent identical letters is the number of $(k-1)$-...
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Counting Lyndon words with no adjacent character repeats

I'm interested in counting aperiodic words in bracelets. I know that corresponds to Lyndon words, and I know how to count the number of Lyndon words for an $(n, k)$ bracelet using Moreau's necklace ...
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How to count necklaces with this additional requirement?

I've recently dug into the idea of necklaces for a project I'm working on, and it's almost exactly what I'm looking for. The way I understand it, the general necklace-counting function is essentially ...
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2 votes
1 answer
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Count of bracelets with no adjacent colours the same

I already know how to calculate the amount of bracelets of length $n$ and $k$ colours. I'd like to add a condition: only count bracelets with no adjacent colours the same. For context, this is a ...
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Count distinct possible words without rotations or reflections

Suppose I have an alphabet ${\{A, B, C, D\}}$ and I want to count all possible words of length $n$. Easy: it's $4^n$. What should I do if I want to count all possible words that: a) are unique given ...
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Terminology of an equivalence class of necklaces given by permuting the alphabet (or "colors")

A necklace is the rotationally-equivalent version of a word on an alphabet. As an example, the two words "$aabaaab$" and "$abaaaba$" are rotations of one another, and represent the same necklace. ...
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4 votes
0 answers
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Necklace infinte sum

Consider the function: $$S(n)=\sum_{j=1}^{\infty}\frac{j^n}{2^j}=\frac1 2+\frac{2^n}{4}+\frac{3^n}{8}+\frac{4^n}{16}+\frac{5^n}{32}+...$$ Euler found the sum of the first few of these as: $S(0)=1$; (...
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Longest run of heads if coins are arranged in a circle?

There are many questions and answers throughout concerning longest run of heads after $n$ flips of a fair coin. For example, this question has multiple detailed answers. A good reference listed was "...
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On the co-primality of bracelet-type numbers

Doing some related research (more info in a past question), I've stumbled upon this interesting problem which I cannot seem to solve myself: Let an integer $N$ be the number of digits imprinted on a ...
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On the co-primality of bracelet-type binary numbers

Let an integer N be the number of digits imprinted on a bracelet, which can come in two values, 1 and 0. You can produce a binary number by writing down the 1's and 0's on the bracelet from left to ...
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2 votes
1 answer
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Number of $n$-bead binary necklaces [OEIS-A000013]

I tried to obtain the number of $n$-bead binary necklaces from my program written in C++. Then, one formula came up when I looked up the number to see if my thought is correct. Number of $n$-bead ...
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Number of Bracelets with 6 white, 3 blue, and 5 red beads

Six identical white beads, three identical blue beads, and five identical red beads are to be strung together to create a bracelet. If the beads are free to move all the way around the bracelet, how ...
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2 votes
2 answers
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Number of different necklaces with $2$ black and $6$ white beads

This is not a home work question, I'm preparing for an entrance test. The number of different necklaces you can form with $2$ black and $6$ white beads is? My approach: We can place the white ...
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Counting Necklaces

Suppose we have a necklace with $n$ beads. Each bead is either red or blue. I'd like to ask how to count the number of necklaces $f(n,m,k)$ satisfying the following requirements: 1) There are exactly ...
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k-aray necklaces with fixed/conserved positions

I asked a question in a previous post about enumerating all possible k-ary bracelets with certain positions fixed to a specific bead/character. I asked a few mathematicians, and it seems its quite ...
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Show COMBINATORIALLY the number of irreducible monic polynomials is the number of primitive necklaces

Show that the number of monic irreducible polynomials of degree $n$ over a finite field of size $q$ is the same as the number of primitive necklaces of size $n$ with $q$ colors. I have the formula $$...
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3 votes
1 answer
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Rotation of necklaces

The number of fixed necklaces of length $n$ with $a$ types of beads is $$N(n,a)=\frac1n\sum_{d|n}\phi(d)a^{n/d}\;.$$ It is clear intuitively that the number of rotational coincidences gets ...
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1 vote
3 answers
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(Music) List of all possible "types of set" of 12 musical notes

I have looked into trying to figure what are all the possible "types" of note set combinations there are and how I would go by listing them if possible. It turns out this is harder than I thought. The ...
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2 votes
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Approximation of products of necklaces

Let's consider approximation of $\prod_{p=1}^n N(p,a)$, $n\to \infty$, where the number of fixed necklaces of length n composed of $a$ types of beads $N(n,a)$: $\frac {(a-1)^{n+1}} {(a-3) \cdot n!} \...
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1 vote
1 answer
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unique bracelets from 6 beads, where 1 bead is a fixed color

I'm trying to calculate the number of unique bracelets of length $6$ that can be made from $11$ differently colored beads. However there is one important key detail! Each bracelet has to contain ...
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3 votes
3 answers
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unique necklace from 6 beads

I'm trying to calculate the number of unique bracelets of length $6$ that can be made from $6$ differently colored beads. A same color bead can be repeated and used as many times as possible as the ...
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2 votes
1 answer
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k-ary bracelets with conserved/fixed indexes

Im using the formula from here: https://en.wikipedia.org/wiki/Necklace_(combinatorics)#Number_of_bracelets to calculate the number of unique bracelets, accepting all rotation/mirroring as identical, ...
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