# 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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### Orientable necklaces and complements

I'm trying to understand the OEIS sequence A059078: Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement. 0, 0, 0, 1, 2, 6, 12, 27, 54,...
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### Necklace with 4p beads (Burnside's lemma)

Let $p \geq 3$ be a prime number. We consider $2p$ black beads and $2p$ blue beads (both indistinguishable). How many unique necklaces of size $4p$, created from these beads, are there? (Consider only ...
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### Individual contribution of string-length counts in polygons drawn on a clock face

Background In a psychology experiment we had people interact with various 5-sided polygons drawn on the face of a clock (the specifics of the experiment are not pertinent to the question at hand). For ...
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### Given one has a cyclic code, how would you deduplicate the other orientations of the codeword in a systematic way?

I've been recently working with Reed-Solomon codes and wanted to make use of their cyclic properties to uniquely identify something regardless of where the reading of the code started symbol-wise. Is ...
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### 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 $n$...
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### Number of colourings of the necklace.

I want to count the number of ways to color beads of a necklace green and red, such that two adjacent beads cannot both be red. The necklace cannot be turned or reflected, the beads are labelled. ...
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### Counting number of bracelet configurations

Suppose we have $3$ red beads and $3$ green beads. How many different bracelets can we make from this? Initially I thought this was a standard circular permutation but I am getting a non integer ...
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### Enumerative combinatorics. Number of unique сhords

Have: Circle N * 2 equidistant points on circle N chords connecting points into pairs Exactly 1 chord connected to each points Additional conditions: Points are equivalent, so a1a2 = a2a1, aabbccdd ...
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### To find the number of ways to choose $3$ vertices up to rotation from $8$ cycle.

Suppose we have a cycle graph on $8$ vertices. To find the number of ways to choose $3$ vertices up to rotation. Note that we can choose $3$ vertices from $8$ vertices in $C(8,3) = 56$ ways. But many ...
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### Different possibilities of arranging perls in a chain using a group action or combinatorial argument

We want to build chains of $6$ perls. The perls are provided in $n$ colors: $n_1, n_2, \dotsc, n_n$. We call two such a chains essentially different, if after a rotation by $180^°$ the chains are ...
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### Algorithm for getting all 2D necklaces

Let's assume that we work in the finite field of two elements, $\{0, 1\}$. We want for example to construct all the necklaces of length $N = 3$. Thus, from the full set of ALL PERMUTATIONS (edit:to be ...
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### 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 ...
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### Arrangements of three types of beads around a circle so that no two beads of the same color are adjacent

Suppose you have beads with colors and numbers on them. There are 8 colored white, 6 colored black, and 3 colored red. The white are numbered 1 to 8, the black numbered 1 to 6, and the red numbered ...
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### Number of different necklaces with $4$ beads, s.t. W$\ge$ B$\ge$ R.

It is stated that to form a 4-beads necklace with white, black, and red beads, s.t. $W \ge B \ge R,$ where the number of white, black, and red beads is denoted by W, B, and R respectively. The text is ...
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Problem: We construct a necklace using 7 green and 9 red beads. How many different necklaces can we construct? Necklaces that can be rotated to each other are considered to be the same. My answer: $\... • 757 8 votes 2 answers 155 views ### Does every sufficiently long string contain consecutive permutations of another string? Let$\mathcal{C}$be a finite set, let$\mathcal{F}(\mathcal{C})$be the free (non-abelian) monoid over$\mathcal{C}$, and let$n\in\mathbb{N}$be an integer. For every$k\in \mathbb{N}$, write$S_k$... 3 votes 2 answers 5k views ### 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 ... • 9,463 2 votes 1 answer 235 views ### 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 ... 1 vote 3 answers 171 views ### 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 ... 3 votes 1 answer 676 views ### 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 ... 1 vote 1 answer 499 views ### 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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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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### 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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