questions regarding the re-orderings of some finite set of objects.
0
votes
1answer
35 views
Explanation of the 'division of $52$ cards' in four groups of $13$ problem?
I have been thinking about this problem, total possible combinations of division of $52$ cards deck in $4$ groups of $13$.
I remember that the answer was ${52 \choose 13 }{39 \choose 13} {26 \choose ...
0
votes
1answer
31 views
Confusing combination-permutations question
In a shop there are five types of ice-creams available. A child buys
six ice-creams. Is it true that the number of different ways the child
can buy six ice creams is equal to the number of ...
1
vote
1answer
71 views
Restricted Permutations and Combinations
Tino, Colin, Candice, Derek, Esther, Mary and Ronald are famous artist. Starting next week, they will take turns to display their work
and each artist's work will be on display at the London Show for
...
0
votes
2answers
55 views
Product of disjoint cycles question.
Consider the following permutations $x$ and $y$ in $S_6$:
$x=(1 \, 3 \, 5)(2 \, 4)$ and $y=(2 \, 3 \, 4 \, 5)$
Express $xy$ as a product of disjoint cycles.
My attempt: I first got $xy = (3 \, 5 \, ...
4
votes
2answers
56 views
How to determine the number of 5 consecutive digit blocks in a set of digits?
Let there be a set containing the following digits: {1,2,3,4,5,6,7,8,9}.
If I choose 5 digit blocks, where the digits are arrange in consecutive ascending order, ...
0
votes
1answer
35 views
The letters A, E, I, P, Q, and R are arranged in a circle. Find the probability that at least 2 vowels are next to one another
This isn't homework, but could someone please give an explanation and answer to this question. Thanks! :D
0
votes
2answers
69 views
The letters A, E, I, P, Q, and R are arranged in a circle. Find the probability that at least 2 vowels are next to one another.
I've had trouble for his one for a while now. All help would be greatly appreciated.
My attempt:
Alright, since one letter is fixed, that leaves us with 5 letters to arrange. I'm going to fix the ...
0
votes
2answers
164 views
Proof of Cayley's theorem. [duplicate]
I have hard time understanding Fraleigh's proof. Can someone either explain Fraleigh's or provide an alternative, perhaps easier proof?
Thanks so much.
3
votes
2answers
80 views
Derangements property
It is not difficult to evaluate a formula for the number of derangements, with a simple combinatorical argument we get $D(n)=(n-1)(D(n-1)+D(n-2)), n\ge 3$ where $D(n)$ is the number of derangements.
...
9
votes
2answers
143 views
Cocktail bar problem
Recently I went out with friends and we asked ourselves the following question: Consider $n$ people sitting at a cocktail bar next to each other. How many rearrangements have to be made to ensure that ...
7
votes
0answers
146 views
Citation for subset complement result
Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
1
vote
2answers
44 views
Permutations in Two Rows
I have been looking at linear and circular permutations. I have now come across a question that entails permutations in two rows.
This is the question:
Six natives and two foreigners are seated in a ...
0
votes
2answers
30 views
Aptitude Permutation Imp Question [closed]
I have one question based on permutation
"Howmany six digit numbers can be formed using digits 1 to 6 without repetition such that the number is divisible by the digit at its unit place?"..Please help ...
0
votes
1answer
41 views
Anti-Symmetric Complex Polynomial
Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have ...
5
votes
4answers
57 views
Permutations under Complex Numbers
The question stands:
Let $S=\mathbb{C}-\{1,0\}$. Describe the subgroup of $\operatorname{Perm}(S)$ generated by the functions:
$f:S\rightarrow S, z\mapsto 1-z$ and $g:S\rightarrow S, z\mapsto ...
2
votes
2answers
42 views
Why cannot the permutation $f^{-1}(1,2,3,5)f$ be even
Please help me to prove that if $f\in S_6$ be arbiotrary permutation so the permutation $f^{-1}(1,2,3,5)f$ cannot be an even permutation.
I am sure there is a small thing I am missing it. Thank you.
4
votes
1answer
35 views
from product of swaps to product of disjoint cycles
I have permutation represented in this form:
$X=(8,9)(14,15)(12,14)(13,15)$
Can I do the following steps?
$$X=(8,9)(14,15)(12,14)(13,15)\\
=(8,9)(14,12,15)(13,15)\\
=(8,9)(15,13,14,12)$$
I think ...
3
votes
2answers
69 views
Calculating the power of permutations
I have this permutation $A$:
$$
\left(\begin{array}{rrrrrrrrrr}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
10 & 8 & 5 & 2 & 3 & 1 & 6 & ...
0
votes
3answers
51 views
Odd permutations example
How to prove that this permutation:
p=(1,10,9,7,6)(2,8,4)(3,5)
is odd.
Thanks much!
3
votes
0answers
51 views
Consider $P(n)$ as a number of $n$-permutations, which each cycle have even length, and …
Consider $P(n)$ as a number of $n$-permutations, which each cycle have even length, and $N(n)$ as a number of $n$-permutations, which each cycle have odd length. Calculate $P(2n)-N(2n)$
3
votes
1answer
65 views
computing a limit of a ratio of derangements
Fix $m$. Consider $\lbrace 1,\ldots ,n\rbrace$. Let $a_1\dots a_n$ be a permutation of this set. How many permutations are there such that $a_i\not=i$ for all $i$ and each $i$ travels at most $m$ ...
1
vote
1answer
62 views
Minimum number of moves to convert a list of any integers into a permutation
Given a list of integers of size n, how to find the minimum number of moves to convert it to a Permutation?
In one move, we are allowed to decrease or increase any element of the list by one.
For ...
0
votes
2answers
99 views
Combinatorics: How to find the number of sets of numbers in increasing order?
The problem is the following one:
Let $n$ and $m$ be natural numbers and $m < n$. Find $m$-permutations of the set $\{1, 2,\dots, n\}$ such that permutations are in non-decreasing order (for both ...
6
votes
2answers
315 views
How many possible iPhone passwords?
A standard iPhone has $10$ digits (ranging from $0$ to $9$) Consider a user who has oily fingers (which is normal for an average user) and he unlocks the iPhone by pressing the numbers on the number ...
0
votes
1answer
28 views
Permutations - bug in Wolfram Alpha?
I've got question whether Wolfram Alpha has bug in showing permutations. Check permutation rules, list, and two-line notation in this permutation.
I'd expect same result as was on Wolfram Alpha blog ...
2
votes
2answers
39 views
number of ways poker card question
I am having difficulties to calculate the number of ways 11 poker card can be chosen such that two cards of one suit, two cards of another suit, four cards of another suit, three cards of another ...
1
vote
2answers
53 views
Notation of permutation
I have a question about the notation of the permutation. I am looking at a proof that shows composition of two permutations is a permutation. There it says, let us assume the contrary, let ...
6
votes
1answer
102 views
Literature on group theory of Rubik's Cube
While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
4
votes
3answers
96 views
Is there a quick trick to write permutations of $S_n$ as products of transpositions?
If I want to write $(123)$ as product of transpositions, I get $(13)(12)$.
For $(132)$ I get $(12)(13)$. For $(1234)$, I get $(14)(13)(12)$. Seems like I can write $(abcd)$ as $(ad)(ac)(ab)$.
Is this ...
0
votes
1answer
44 views
Is there a known distribution for this permutation with replacement problem? [duplicate]
Choose $t$ numbers from $n$ $(n>t)$ distinct numbers with replacement and the order of the $t$ numbers matters.
Say,
$P(X=1) = \dfrac{{numbers\ of\ unique\ t-set \ which\ has\ 1\ distinct\ ...
2
votes
2answers
138 views
Solve for numbers to appear on two six-sided dice
I have a small wooden calendar that uses two six-sided dice to display day of month. One die carries numbers 0, 1, 2, 6, 7, 8 and the other carries 0, 1, 2, 3, 4, 5. The six of course doubles as the ...
0
votes
2answers
33 views
Relation between binomials
how can I prove that the following relation is true:
$$\binom{x-2}y+2\binom{x-2}{y-1}+\binom{x-2}{y-2}=\binom{x}y$$
Thank you for hints or references!
Marted
2
votes
1answer
52 views
Number of Permutations Fixed by the Fundamental Transformation is Fibonacci
Writing a permutation in $S_n$ as a product of disjoint cycles, we define a standard representation by writing each cycle with its largest element first, and ordering the cycles by the increasing ...
4
votes
1answer
62 views
Finding a permutation, and number of, from powers of the permutation
Sorry for the vagueness of the title, I couldn't think of a better way to put it. I just wanted to run a couple of simple questions past SE to check my reasoning is correct etc.
Find a permutation ...
2
votes
1answer
48 views
Algebra, groups and permutations
The question asks for me to write down the permutations on the set $\{1,2,3,4\}$ which are symmetries of the square with vertices as shown. Hence show that $D_4$ is a subgroup of $S_4$.
1 2
4 ...
1
vote
4answers
55 views
Combinations of consecutive digits
Find the number of passwords that use 3, 4, 5, 6, 7, 8, 9 exactly once.
I think I solved this part: it's 7!
Next question is: in how many of those 7! are the three even digits consecutive?
I been ...
1
vote
2answers
34 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 that ...
0
votes
1answer
52 views
Number of $k$-cycles in permutations of $[2k]$?
What is the expectation of the number of $k$-cycles in a randomly selected permutation of $[2k] = {1,2, . . . ,2k}$?
0
votes
2answers
147 views
A step in finding the determinant of transpose of a matrix
The following question involves the permutation group, which I am horrible at handling. Any help will be greatly appreciated.
Let $A$ be an $n \times n$ matrix with entries $(a_{ij})_{i = 1,2 \cdots, ...
1
vote
0answers
29 views
Permutation and combination with males/females
There are $6$ males and $6$ females in the finals of a talent competition. A contest is held to pick the top $3$ winners in both the male and female categories in order of merit. How many different ...
2
votes
2answers
104 views
Combinatorics And Music [closed]
What's the likelihood that we'll run out of different songs to be able to make?
I know this seems like a difficult question to answer, possibly. But I was just wondering, and I thought of a scale of ...
2
votes
3answers
85 views
Stanley's Enumerative Combinatorics Question help
This is a question from supplement( Bijective proof problems ) to the Stanley's Enumerative Combinatorics.
The question statement goes like this.
"In how many ways can $n$ square envelopes of ...
1
vote
2answers
117 views
Combinatorics: counting sums with conditions
Hi guys,
Here's a combinatorial nut to crack. I've been struggling with this one:
Count the number of ways summing a set of $n$ non negative integers $i_1, \cdots, i_n \in \{ 0, \cdots , n-1\} $ so ...
0
votes
1answer
24 views
Of 15 marbles 1 is black 6 are white and the rest some other color
How many ways can they be arranged so the black marble will not be next to a white marble
3
votes
1answer
100 views
What is the number of even and odd permutations that satisfies the following condition?
Let $\phi$ be a permutation of $n$ numbers with $\phi(1)=1$ and $\phi(2)=2$. It is asked to prove that the number of odd permutations of $n$ numbers that commute with $\phi$ is equal to the number of ...
1
vote
1answer
57 views
Equation to find possible combinations of combining lists while retaining order
Lets say I have two ordered lists of size n, [A1, A2, ..., An] and [B1, B2, ...,Bn]. I want to find all the possible ...
6
votes
1answer
81 views
On the precise asymptotic scaling of $n!/(n-k)!$ as $n,k \to \infty$
On page 23 of [Erdős+Rényi 1960, "On the evolution of random graphs"], the following asymptotic formula is stated without proof:
$$ \binom{n}{k} \sim \frac{n^k \mathrm e^{-\frac{k^2}{2n} - ...
1
vote
0answers
140 views
Possible paths in 3×8 GRID (from Brilliant.org)
Let G be a rectangular grid of unit squares with 3 rows and 8 columns. How many self-avoiding walks are there from the bottom left square of G to the top left square of G?
A self-avoiding walk on a ...
3
votes
3answers
86 views
Showing that a transitive abelian permutation group is necessarily regular
I am trying to show that a transitive, abelian permutation group acting on a set $X$ is necessarily regular, given this hint: 'Given $g \in G$, consider the set $X^g:=\{x \in X\,|\,gx=x\}$. Show that ...
1
vote
1answer
41 views
The set that a group can act on it as a permutation group
What can we say about the set that a group can act on it as a permutation group? My question is about the structure of elements of a group G when acts on an arbitrary set Ω as a permutation group. For ...
