Questions tagged [permutations]

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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Confusion over Combinations and Permutations

Just when I thought I understood everything, I have yet again made myself confused and cannot resolve this issue. Consider selecting 3 people from 5 where the order of selection matters, this is ...
James Chadwick's user avatar
-2 votes
1 answer
44 views

Does the Symmetric Group on 71 letters get specific attention by researchers in simple groups? [closed]

Since it contains all the sporadic groups and no smaller symmetric group does, it might be of special importance.
Richard Peterson's user avatar
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0 answers
12 views

Selecting subsets $G$ of a set $\mathcal{K}$ of integers so that atleast one subset of $G$ has consecutive integers in wrap around sense.

Let $\mathcal{K}=[1,2,\cdots,K]$ be a set of cardinality $K$. For parameters $a$ and $p$, taking integer values, how many subsets $G$ of $\mathcal{K}$ exist of cardinality $(1+a+p)$ such that there is ...
WorkingFisherman's user avatar
0 votes
0 answers
11 views

randomized left stochastic matrices: two by two right multiplied by two by one as an average

I'm looking at the identity matrix and its inverse right multiplied by two by one matrix [1,0] and two by one matrix [0,1]. Choosing either is a fifty fifty chance. If I simply right multiply and ...
floor cat's user avatar
  • 161
1 vote
0 answers
25 views

Sums of characters over over partitions of equal length

Let $\chi^{\lambda}$ and $\chi^{\mu}$ be irreducible characters of the symmetric group $S_n$. Their inner product satisfies $\langle \chi^{\lambda}, \chi^{\mu}\rangle =\sum_{\nu} \frac{1}{z_{\nu}} \...
Andrew's user avatar
  • 551
3 votes
0 answers
41 views

Sorting a permutation by sorting half of the elements at a time

The problem: suppose we have an array $p[1..n]$ (where $n$ is a multiple of 4) which initially contains some permutation of the numbers $\{1, 2, \ldots, n\}$. The only way we are allowed to modify ...
janezb's user avatar
  • 31
0 votes
0 answers
19 views

Restore shuffled function output

I have a function $\mathbf{y} = f(\mathbf{x})$ that takes a vector $\mathbf{x}$ as input and outputs another vector $\mathbf{y}$. The output is perturbed as follows: $\tilde{\mathbf{y}} = \pi \cdot \...
Zhengyi Li's user avatar
1 vote
1 answer
78 views

Number of ways to arrange characters in the alphabet [closed]

Given an alphabet of size A and a bowl of alphabet soup of size S. Assuming that the distribution of characters in any one bowl is uniform, what is the likelihood that a random bowl contains at least ...
Maximilian's user avatar
0 votes
0 answers
9 views

Exchangeability in terms of permutations in card theory, Ethier (2010).

Exchangeability in terms of permutations in card theory, Ethier (2010). I am experiencing doubt with reconciling the standard definition of exchangeability with the way Ethier (2010), in Doctrine of ...
microhaus's user avatar
  • 924
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0 answers
27 views

In how many different ways can we color 12 balls into red, green and yellow? [closed]

So basically, I'm stuck between two methods, and I don't know which one is correct. Am I supposed to solve it like this 12!/3!9! = 1211109!/3!*9! = 1320/6 = 220 differnt ways or. Am I stupid and its ...
Shayan Bajwa's user avatar
-3 votes
2 answers
67 views

What is the probability that one simultaneous roll of five dice gets a four? [closed]

When playing Yahtzee!, five dice are rolled simultaneously. What is the probability that one roll of five dice gets a four. That is that four of the dice each show k st dots at the same time as a die ...
First_1st's user avatar
-3 votes
1 answer
29 views

Transposition $q(j)$ equal to permutation $p(i)$? [closed]

The book tells me that, for any permutation $p$ and a transposition $q=p.t$, $q(i)=p(i)$ Now, the basic transposition is expressed as: $$ \sigma \begin{pmatrix} j&i\\\\ i&j \end{pmatrix} $$ So,...
MonkeyDL's user avatar
0 votes
1 answer
26 views

How to find different representation of permutation in product of transpositions?

It is known that product of transposition to write is not unique. it can be written in many ways,there is theorem that stats that all representation of product of transposition for a given cycle are ...
kaushal trada's user avatar
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0 answers
34 views

Computation of the symmetric number of a finite group [duplicate]

I got a bit curious about the concept of the symmetric number of a finite group, and decided to do some computations with GAP to determine their values for some small finite groups. The symmetric ...
Justin Benfield's user avatar
1 vote
0 answers
43 views

Maximizing the number of colors so that every subgrid contains all colors

Consider an $n\times n$ grid. Define the set $S$ as subgrids shapes which includes all $(i,j)$ pairs so that $i\times j=n$. eg: we can take $i=1, j=n$ which is a row shape structure and it belongs to ...
Happypantsdw's user avatar
3 votes
1 answer
54 views

Number of arrangements of "AABCXX" without "AA" and "C" isn't next to "A" or "B".

I'm trying to use the Goulden-Jackson cluster method with the alphabet $V=\{A,B,C,X\}$ and forbidden bad words $\{ AA, AC, CA, BC, CB\}$. I also want to keep track of how many of each letter there are,...
ploosu2's user avatar
  • 8,886
1 vote
0 answers
33 views

A Natural Probability Distribution on the Infinite Symmetric Group

Is there a "natural" probability distribution on the set of bijections from $\mathbb{N}$ to itself? Preferably, I would want a distribution which arises from some combinatorial procedure. ...
Miles Gould's user avatar
1 vote
1 answer
50 views

Permutations Doubt: How many $5$-digit whole numbers with no $0$s are divisible by $6$?

I came across this combinatorics question that was quite interesting, as in methods to solve this. How many $5$-digit whole numbers with no $0$s are divisible by $6$? First, I found the number of $5$-...
lightningjay's user avatar
4 votes
1 answer
118 views

Permutations of 10 players within 2 Badminton courts: Covering $10$-vertex complete graph $K_{10} $ by two disjoint $K_4$

I am facing this everyday problem and I wanted to actually see how to formalise and reason on. We have 10 players and two courts in our badminton matches. We define a shift to be an instance of ...
Ramit's user avatar
  • 125
2 votes
2 answers
47 views

In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random

In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random. From each game a winner is decided on the basis of a game played between the two players of the ...
mathophile's user avatar
  • 3,513
1 vote
0 answers
112 views
+150

Can we do any better bijective mapping of a permutation series which is only bijective for a probabilistic subset of it's input domain

So we want to bijectively map one path to another. But depending on start and target node we can only choose from a subset of all transitions. It would look like this: We also do not know where one ...
J. Doe's user avatar
  • 67
1 vote
0 answers
34 views

permutation of the determinant according to the groups $S_{n1}$ and $S_{n2}$?

Reading about alternating linear n maps, I found this alternative definition of a determinant, based on its permutation expression (which iteratively sums the product of all the permutations that can ...
MonkeyDL's user avatar
0 votes
0 answers
34 views

Understanding the generalized "Birthday Problem" formula [duplicate]

While practicing frequently asked probability questions during interviews, I came across the classical "Birthday problem". While I understand some of the reasoning explained on wikipedia, ...
Julien Maas's user avatar
-2 votes
0 answers
20 views

Combinatorics question related to stars andbars #permutation #combination [closed]

A community with n members chooses its representative by voting. a) In how many ways can “open” voting result, if everybody votes for one per- son (perhaps for himself/herself)? Open voting means that ...
user1311073's user avatar
4 votes
3 answers
351 views

Number of ways in which 5 girls and 5 boys can be arranged in a line such that only 4 girls stand adjacent to one another

I have already calculated n but i am confused to find the value of m. I have tried the following process:- Selected 4 girls from 5 girls in ${}_5C_1$ ways and treated it as a unit After selecting 4 ...
Dhanuj Pathak's user avatar
-1 votes
2 answers
32 views

combinations problem - teachers to groups [closed]

My question is in how many ways can you divide 20 teachers into two groups, one group would have 15 people and the other would have 5. I tried Benjamin Dickman's formula but it did not work out =, the ...
Aaryan Velluri's user avatar
2 votes
0 answers
43 views

Number of all possible arrangements such that no two objects of same kind are together

There are $k_1+k_2+k_3+...+k_n$ objects of which $k_1$ are of first kind, $k_2$ are of second kind, $k_3$ are of third kind, ..., $k_n$ are of $n^{th}$ kind. Calculate the number of all possible ...
Sparsh Gupta's user avatar
0 votes
2 answers
60 views

Number of solutions to $x+y+z=6$, where $x,y,z \in [0,5] \cap \mathbb{Z}$ and are distinct

My method: Given that $x,y,z$ are distinct. Let $x<y<z$, $a = x$, $b = y-x$, and $c = z-y$. So the question transforms into finding $a,b,c$ such that $3a+2b+c = 14$, $a,b,c \in [0,5]\cap\mathbb{...
Maths lover's user avatar
0 votes
0 answers
20 views

Total number of ways to paint a cube of six colours [duplicate]

I came across this question recently but couldn't satisfy myself with a good explanation. Please provide an explanation The number of distinct ways of painting six faces of a cube with six different ...
T J's user avatar
  • 1
0 votes
1 answer
48 views

Question 7 from A first course in Probability by Sheldon Ross 10 ed. chapter 2.

A bus departs from Bus Stop 16 with 20 passengers. The bus's journey will end at Bus Stop 20. Passengers can leave the bus at any stop, but no passengers can board. Work out the number of ...
Abhishek Singh's user avatar
0 votes
0 answers
16 views

How to interpret this sequence of tuples?

I'm not sure if I understand the following sequence of tuples $\{(x_k,A_k),(y_k,B_k)\}^n_{k=1}$, where $x_k \in A_k$ and $y_k \in B_k$, $(y_k)^n_{k=1}$ is a permutation of $(x_k)^n_{k=1}$, and $(B_k)^...
Hans Brecker's user avatar
1 vote
4 answers
156 views

How to find the number of elements in $S_7$ that commutes with $(123) (245) (456)$?

The following question was asked in my masters entrance exam and I would need help in finding the correct number. Question: The number of elements in $S_7$ that commutes with $(123) (245)(456)$ is ......
BKFH's user avatar
  • 124
7 votes
1 answer
142 views

permutation group on $\mathbb{F}_p$

Suppose $\mathbb{F}_p$ the field with $p$ elements and $\mathcal{S}(\mathbb{F}_p)$ the permutation group. Define for $a \in \mathbb{F}^\times_p$ and $b \in \mathbb{F}_p$ the permutation $\gamma_{a,b}: ...
riescharlison's user avatar
3 votes
1 answer
45 views

'$n\times p$' passengers in '$n$' identical boats (which can accommodate '$p$' passengers each)

There are $n$ identical boats and each boat can accommodate $p$ passengers. What are the total number of ways $n\cdot p$ passengers can be accommodated in $n$ boats? Here's my approach to solve the ...
Ishant Dumane's user avatar
2 votes
1 answer
85 views

Why does the number of permutations of these sequences with non-negative partial sums have such a simple closed form (m choose n)?

I've been thinking about a problem, and I think that I have a solution, and I'm not sure why it works. Looking for an intuitive (or just any) explanation. The problem Choose an integer $k>1$. For ...
Quick_Fix's user avatar
1 vote
1 answer
55 views

Derangements for couples in a round table

Question: Let ( m(n) ) denote the number of ways of seating ( n ) married couples around a circle such that no husband sits next to his wife. Then, the remainder obtained on dividing ( m(5) ) by ( 5 ) ...
OpateItZOpatoOpate's user avatar
1 vote
1 answer
55 views

The number of lattice paths from (0, 0) to (2n, 3n) that never go above the diagonal

In discrete mathematics, a specific type of problem focuses on the number of lattice paths which go from one point A to another point B $\textbf{WITHOUT}$ going above the diagonal in the 2D plane. For ...
grj040803's user avatar
0 votes
1 answer
23 views

Why can we treat samples/permutations as happening one by one?

When considering a sample over some set, we often picture picking elements for the sample one by one. This is used implicitly in many problems. For example, in determining how many $k$-length samples ...
araj's user avatar
  • 3
0 votes
1 answer
53 views

Total permutations for n-married couples

Let A(n) denote the number of ways of seating n-married couples, around a circle, such that men and women sit alternately, and no husband sits next to his wife. Then Compute A(5): I tried applying ...
OpateItZOpatoOpate's user avatar
1 vote
1 answer
42 views

Proving Row Independence in Matrix Product Resulting from Linearly Independent Vectors and Permutations

Let's denote by $u_1, u_2, \ldots, u_n$ a set of linearly independent vectors. For each $i$ from $1$ to $n$, we define $\sigma_i$ as distinct permutations of the numbers $1$ to $n$. Construct matrix $...
Fernand's user avatar
  • 15
0 votes
0 answers
22 views

Most efficient way to recompute matrix cost after swapping columns

I work with regular matrices $M \in \mathbb N^{n\times n}$ with $n > 1$. My cost function is: $$\text{Cost} = \sum_{i=1, j>i}^n M_{ij}$$ Hence, we only consider the superior triangle, diagonal ...
Albert Schrödinberg's user avatar
1 vote
0 answers
24 views

A pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ such that $f$ and $g$ are idempotent, commute with each other and $f \times g$ is bijective

The question The question is: Does there exist a pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ satisfying the following properties? $f$ and $g$ are idempotent, meaning that $\forall n \in \...
Smiley1000's user avatar
  • 1,025
2 votes
0 answers
33 views

Possible differences of integer sequences and their permutations

Given a sequence $(a_n)_{n \in \mathbb{N}}$, what are necessary and sufficient conditions on $(a_n)_{n \in \mathbb{N}}$ so that there exists a sequence $(b_n)_{n \in \mathbb{N}}$ and a bijection $\...
Smiley1000's user avatar
  • 1,025
3 votes
2 answers
93 views

Counting $10$ length paths in a $2 \times 4$ rectangle with distance $6$ units from start to end meaning negative moves allowed?

How many different routes of length 10 units (each side is 1 unit) are there to traverse from lower left corner (point A) to top right corner (point B) in a rectangle with 2 rows and 4 column cells ...
Jonny Boy1's user avatar
-1 votes
0 answers
24 views

I'm trying to create a match schedule for pickleball

I have two teams each with 12 members. Matches will be doubles play. I want each member to play only once with each teammate. That's a total of 11 games each, 66 in total. I want each player to play ...
Richard Appleby's user avatar
4 votes
1 answer
116 views

Formalising the problem and create a proof for the game "Waffle"

Waffle is an online game at https://wafflegame.net/daily. It consists in moving letters (swapping them) to recreate the original words. While you have 15 moves, it can be done in 10. I usually try to ...
user's user avatar
  • 1,123
0 votes
1 answer
86 views

How to determine what is n and what is r?

In case of permutation with repetition, we have formula = $n^r$ How do you decide which thing will be $n$ and which will be $r$? Like in this question: Your mother-in-law buys 1000 small gifts to give ...
Raj Ishu's user avatar
1 vote
1 answer
54 views

Combinations and forming groups [closed]

In how many ways can three groups of 3 be made from letters of the word CROCODILE? PS: Answer is 140 but I fail to see the logic. My take was (9C3 *6C3 *3C3) / (3! *2! *2!) =70 but its wrong
AMathsStudent's user avatar
0 votes
0 answers
32 views

There is a permutation matrix $P$ such that $PAP^{T}$ is in this form for symmetric $A$

Suppose that $A$ is a real matrix and is symmetric and nonzero, then I want to prove that there is a permutation matrix $P$ such that $PAP^{T}=\left[\begin{array}{ll} B & E^{\top} \\ E & C \...
YuerCauchy's user avatar
1 vote
2 answers
61 views

I have a doubt for a combinatorics question below and want to know where am I going wrong.

In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that ...
shubham kakade's user avatar

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