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

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2
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1answer
389 views

How to assign values to letters to create unique values per word when all letters are added together?

I'm writing a program to match anagrams in order to practice coding. One way I want to try this is to assign values to letters such that adding up the letters in the individual words creates a unique ...
1
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2answers
115 views

Assigning values to permutations

$N$ objects can be arranged in $N!$ different orders. For example, $10$ playing cards can be stacked $10! = 3,628,800$ different ways. Is there a way to assign a numerical value to each permutation so ...
1
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2answers
68 views

Showing that a quotient group $G/N$ is isomorphic to $\mathbb{Z}_3$

I have permutations $\sigma=(135)(27)$, and $\tau = (27)(468)$. $G =\langle \sigma,\tau \rangle$ and $N$ is the smallest subgroup of $G$ that contains $\tau$, so $N = \langle \tau \rangle$. $|\sigma| ...
5
votes
3answers
256 views

How many 6 digit numbers are possible with no digit appearing more than thrice?

How many 6 digit numbers are possible with at most three digits repeated? My attempt: The possibilities are: A)(3,2,1) One set of three repeated digit, another set of two repeated digit and ...
2
votes
1answer
21 views

Expressing a permuation as a product of disjoint cycles.

The theorem: Let $p$ be a permutation of $\{1,\ldots,n\}$. Then $p$ can be expressed as a product of disjoint cycles. How would you express a permutation that permutes every element of the set, ...
0
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1answer
34 views

Number of ways to arrange the alphabet, A is not in first position, B is not in second, and so on.

My first answer was 25! as The first letter has 25 possibilities, second letter 24 possibilities, and so on. However, I realised that it is actually more than that. There is a possibility that the ...
2
votes
1answer
86 views

Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $). Looking at 3-digit ...
0
votes
1answer
393 views

Determining Permutation/Combinations with Bit Strings

I've got a discrete math problem on my hands...I'm trying to understand the steps behind working with bit strings; specifically, how a bit string of x length has "at least" or "exactly" a certain ...
1
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0answers
32 views

Sum of some number of prime numbers is n. What is the upper bound on their product?

Sum of some number of prime numbers is n. What is the upper bound on their LCM (Or you can assume that all primes making up sum n are distinct)? I got some bound esqrt(n * ln(n)). Is there some ...
2
votes
5answers
148 views

proving that $\frac{(n^2)!}{(n!)^n}$ is an integer

How to prove that $$\frac{(n^2)!}{(n!)^n}$$ is always a positive integer when n is also a positive integer. NOTE i want to prove it without induction. I just cancelled $n!$ and split term which are ...
0
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2answers
39 views

$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. ...
4
votes
3answers
84 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
1
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2answers
33 views

In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together?

In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together? My attempt: $5$ student can sit $(5-1)!$ in round table. A teacher can sit between ...
0
votes
3answers
33 views

How many numbers of $7$ digits can be formed with the digit $0,1,1,5,6,6,6$.

How many numbers of $7$ digits can be formed with the digit $0,1,1,5,6,6,6$. My attempt: Seventh place, total number of possibility is $=\frac{6!}{2!\times 3!}=60$ ways. Sixth place, total ...
2
votes
1answer
28 views

How many $3$ digit different number that will be divisible by $5$ can be formed from the digit $0,2,3,4,5,6$ lying between $100$ and $1000$.

How many $3$ digit different number that will be divisible by $5$ can be formed from the digit $0,2,3,4,5,6$ lying between $100$ and $1000$. My attempt: Divisible by $5$ is possible only when ...
0
votes
1answer
16 views

Demonstrate that the product of the permutations(regardless of order) of $S_4$ is not equal to $a$

$S_4$ is the set of all permutations of length 4 Let $a=\binom{1\,2\,3\,4}{3\,2\,1\,4}$ I found that $a$ is an odd permutation and I want to demonstrate that the product of the permutations is even ...
0
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0answers
29 views

Constraint for matrix representation for general irreducible permutation group.

Say I have a matrix $\bf P$ for which is ensured that $P_{ij} \in \{0,1\}$. Then consider this requirement: $$\sum_{k=0}^{n-1}{\bf P}^k[1,0,\dots,0]^T = [1,1,\dots,1]$$ Should this be enough to make ...
1
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2answers
57 views

Problems on Derangements - Combinatorics [on hold]

Determine the number of permutations of $ 1,2, \ldots ,8$ in which no even integer is in its natural position. Please solve this using concept of derangements.
1
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1answer
399 views

Circular Nonconsecutive Permutations

A carousel has eight seats, each representing a different animal. Eight girls are seated on the carousel facing forward (each girl looks at another girl's back). In how many ways can the girls change ...
3
votes
1answer
29 views

the MISSISSIPPI problem - 5 letter word and 6 letter word with constraints

1) Number of ways of selecting 5 letters such that 3 are alike of one type and 2 are alike of another type. There are only 2 ways of selecting 3 letters of the same type i.e., III or SSS. Now with ...
2
votes
3answers
28 views

Permutation and order

I know that the order of a group is the number of the elements, then if we have a permutation what does the order of permutation mean? The number of distinct elements?
1
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0answers
29 views

In how many ways consonants and vowels alternatively for letters of word `CONSTITUTION`. [on hold]

In how many ways consonants and vowels alternatively for letters of word CONSTITUTION. My attempt: The word 'CONSTITUTION' has 7 consonants (C N S T T T ...
0
votes
0answers
25 views

Find $T_1(\langle (1,2,3,4,5,6,7,8,9) \rangle )$ [on hold]

$T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ I am unsure what to do. Let that long permutation be $b$. Do we just find calculations of $b$ like $b^2$ or $b^{-1}$ or $b^b$ etc, that would give an ...
1
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2answers
38 views

How can I simplify this number theory problem?

Let X = {1, 2, 3, 4, 5, 6} and σ= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 3 & 5 & 2 & 1 \\ \end{bmatrix} Define a relation ∼ on X as ...
0
votes
1answer
26 views

How do we solve these permutation and combination questions? [on hold]

Q1 In how many ways a panel of six doctors is selected from five surgeons and six physicians if condition is surgeons are more than physicians. A 82 B 81 C 65 D 135 Q2 Find the no. of ...
0
votes
1answer
453 views

binary strings and counting sequence problem

Hello i'm working on these questions and I have few questions 1) A binary string is a finite sequence of 0 and 1. Ex. 001101 is a string of length 6 a) List all binary strings of length 4 (so I ...
0
votes
1answer
30 views

Finding maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
1
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3answers
32 views

How to find the Permutation in S8 given as the product C= (1483)(12765)(34687)?

I don't understand how to answer this. Just by reading it off, "1 goes to 4, and 4 goes to 6, thus 1 goes to 6" but that logic doesn't match with the answer i've been giving: Answer: (127)(386)(45). ...
1
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0answers
12 views

Maximize the mutual permutation disparity

I am trying to work on a problem that needs me to find the top-k most disparate permutations for a n-tuple (hence n! possible choices). The disparity measure between two permutations I'm thinking of ...
0
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0answers
19 views

$A^{\pi}$ property

Can someone give an example for the matrix mentioned in the following definition. Definition is taken from "Iterative Solution Methods" of Owe Axelsson. Definition: The matrix $A$ said to have ...
0
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0answers
18 views

How do I calculate such possible number of total and serial schedule?

Consider the following two transactions $T_1$ and $T_2:$ How many non serial schedules are possible, if we execute both transactions concurrently? $3000$ $3001$ $3002$ $3003$ My try: ...
-2
votes
1answer
31 views

Simply showing the addition of permutations

How can I show for example AB+BC+AC simply. It is adding up the permutations of n numbers. Another example would be ABC+ABD+ACD+BCD. Sorry I'll try to make it clear with an example ( which is sort of ...
1
vote
2answers
60 views

number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$

Let $A=\{1,2,3,4\}\;,$ Then total number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$ $\bf{My\; Try::}$ If $f(x)=x\;,$ Then $f(f(x))=x.$ So there are ...
0
votes
2answers
30 views

3 men have 4 coats , 5 waistcoats and 6 caps. Then in how many ways can they wear them?

The question is in the title itself. First, I would like to share how I solved this problem at first: We have $4$ coats, $5$ waistcoats and $6$ caps. So, I considered that each man wears one coat, ...
0
votes
2answers
37 views

100 shoelaces, pick 2 random ends and tie them together, what is the probability that a loop is created?

The question is: There are 100 shoelaces in a box. You pick two random ends and tie them together. Either this results in a longer shoelace (if the two ends came from different pieces), or it ...
2
votes
1answer
49 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
1
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1answer
15 views

Number of recursive permutations of all sizes

Consider you have a set of $n$ elements. Now, create all the possible permutations of $k$ elements. Finally, for each permutation create all the possible combinations with the permutations of the ...
0
votes
1answer
42 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
4
votes
1answer
39 views

How many 4 digit pins on set {0-9}

A password can be any 4 digit {0...9}. 1.)How many possible passwords are there? for this I did $10^4 = 10,000$ 2.) How many possible passwords with no repeated digits? $10*9*8*7 = 5040$ 3.) How ...
0
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0answers
31 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$. [closed]

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
13
votes
5answers
4k views

A4 has no subgroup of order 6

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = ...
0
votes
2answers
29 views

4-Sequences {0…9}

My questions are given the set {0,1,2,3,4,5,6,7,8,9}, 1) How many 4-sequences are there? (would this be $10*10 * 10 * 10 = 10,000)? $ since the max possible numbers given to each 4 slots is 10. 2) ...
1
vote
1answer
27 views

5 letter password either lowercase or uppercase

Given that you can have 5 letter password that contains either lowercase or uppercase. My questions are: 1) How many possible passwords are there? I did $52^5 = 380,204,032$ since there are 52 ...
0
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0answers
22 views

Combination and Permutation How many words can be formed? [duplicate]

A contest consists of finding all code words that can be formed from the letters in the word "alpha".Assume that the letter "a" can be used twice but the others at most once: a)How many five-letter ...
0
votes
1answer
15 views

Product of permutations and subgroup generated by permutation(s)

I'm get confused while working with permutations, so I have some questions. $\sigma$ = (1,7,3)(2,10,4,8) $\rho$ = (3,7) $\tau$ = (1,7) First I am told to compute $\tau$$\sigma$$\tau^{-1}$ I dont ...
1
vote
2answers
69 views

Count the permutations which are products of exactly two disjoint cycles.

Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$ such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$. Calculating $a_4$: Possible ...
-1
votes
3answers
63 views

Number of permutations which are products of exactly two disjoint cycles. [duplicate]

Let $l_{n}$ denote the number of those permutations $f$ on the set $A=\{1,2,....,n\}$ such that $f$ is the product of exactly two disjoint cycles. Show that $l_{5}=50.$ I tried a lot but reached ...
0
votes
1answer
16 views

Odds of an event happening

Trying to get my head around the correct way of approach this. You are able to use any letter of the alphabet or number allowing for 36 options, with this you are to create PIN of length 4, for ...
0
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0answers
33 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: ...
3
votes
1answer
31 views

Possible ranks of a $n!\times n$ matrix with permuted rows

Let $a_1,\cdots,a_n$ be $n$ arbitrary real numbers. Form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the $n$ numbers given. Find all the possible ranks of such a matrix. ...