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

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12
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405 views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
8
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0answers
170 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
8
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0answers
205 views

How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. ...
7
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0answers
127 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
7
votes
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193 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 ...
6
votes
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106 views

Knuth equivalence of unshuffles

For any permutation $\sigma\in S_k$ and any $i\in\left\lbrace 1,2,...,k\right\rbrace$, I will denote $\sigma\left(i\right)$ by $\sigma_i$. For any word $w$, I will denote by $\mathrm{st}w$ the ...
6
votes
0answers
63 views

functions representable as a sum of two permutations

I am trying to prove that for every function $f:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ satisfying $\sum_if(i)=0$, there exist permutations $\pi_1, \pi_2:\mathbb{Z}/n\mathbb{Z}\to ...
5
votes
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38 views

Number of representatives from states to from a comittee?

Among the three representatives to a conference from each of the fifty states, either none, one, or two of the representatives will be chosen for a large special committee. How many ways can this ...
5
votes
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198 views

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
5
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48 views

How to “see at a glance” the solution to the exercise “Show that $\langle (1,2,3… n),(1,2,3… m)\rangle$ contains a 3 cycle, if $1 < n < m$”?

I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to ...
5
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0answers
61 views

largest permutation group of odd order

For general $n$, what is the largest subgroup of the symmetric group $S_n$ that has odd order? I have a feeling that it might be the Sylow 3-subgroup...ADDED: but it isn't, as Mark Bennet points out ...
5
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207 views

Special Products of Transpositions

[Edit. Significantly expanded to add examples and (I hope) clarification. Feel free to skim by reading the gray boxes.] A colleague asked me for insights on a collection of special permutations, ...
5
votes
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135 views

How find this $aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$

Question let $m$ is positive numbers,and such $m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^{m-1}+2\times (m+1)^{m-2}+\cdots+(m-1)\times (m+1)+m$$(or see ...
4
votes
0answers
84 views

Cayley's theorem — more than one isomorphism

I've just been learning about Cayley's theorem and a couple of things occurred to me: We know that every finite group of order $n$ is isomorphic to some subgroup of $S_n$. But perhaps there are ...
4
votes
0answers
51 views

Permutation order statistics integral

Let $U_i$ be $[0,1]$ i.i.d. uniform random variables, for $i=1,\ldots,n$. As an example, let $n=3$. Now pick an ordering, say $x_1>x_2<x_3$. and consider the order statistics integral ...
4
votes
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88 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
4
votes
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103 views

Number of ways to order a list of permutations by swapping

I want to solve a problem and I have no idea how or where to begin. I don't even know if it's possible to solve. I tried to find any clues in books about discrete maths but I didn't find anything that ...
4
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0answers
116 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
4
votes
0answers
85 views

Sign of some permutation.

I trying to find the sign of this permutation: $\left(\begin{array}{cccccccccc} 1 & 2 & 3 & \cdots & \cdots & \cdots & \cdots & \cdots & n-1 & n\\ 2 & 4 & ...
4
votes
0answers
96 views

Permutation: How many ways to put 7 people in 10 rooms?

How many ways can 7 people be placed into 10 rooms, if (only) 2 of them can’t share a room with anyone? I'm not sure how to go about this, mostly because of the "share a room" bit. I'm thinking I ...
4
votes
0answers
75 views

Selecting k numbers out of N sorted numbers to a minimize a condition/Formula

A sorted list of $\mathbf N$ numbers is given. $X_1$ $\le$ $X_2$ $\le$ $X_3$ $\le$ .... $\le$ $X_N$ Select $\mathbf K$ Numbers - $Y_1$ , $Y_2$ , $Y_3$ , ..... , $Y_K$ - Such that the following ...
4
votes
0answers
124 views

Is there a name for such matrices

Let $Z$ be a $K \times K$ matrix. All its left diagonal elements are zero. Further it satisfies the following properties: 1.) $Z[ i ][ j ] = Z[j][i] > 0$ for $1 ≤ i < j ≤ K$. 2.) $Z[i][j]$ ...
4
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45 views

Classification of $n \times n$ matrix in which each two components differ in each row and column up to automorphism

Suppose, we define a class $A$ of $n \times n$ matrix as follows: $$\text{In each Row }i, \text{for any} j,k\ (1 \leq j,k \leq n )\ a_{ij} \neq a_{ik} $$ $$\text{In each Column }l, \text{for any } ...
4
votes
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73 views

Expectation of Reciprocals Involving Permutations

Let $a_i$ be $n$ distinct real numbers. What is the expectation: $$\mathbb E_\sigma \left[ \sum_{i=1}^{n} \frac {1} {a_{\sigma(i)} - \sum_{j=1}^{i-1}a_{\sigma(j)}} \right] $$ where the expectation ...
3
votes
0answers
76 views

Number of $n$-permutations for which ${\tau}^k = id$

I am curious about the formula(any closed form) for the number of $n$-permutations $\tau$ such that ${\tau}^{n-1} = id$. How about for the case ${\tau}^n = id$ ?
3
votes
0answers
60 views

Is there away to swap adjacent items in a sequence, such that each permutation occurs once?

Let's us say you have a finite sequence of things. Some are identical, some distinct. For example: $$\langle 2,5,7,2,3\rangle$$ Now, under what conditions can each permutation of a sequence be ...
3
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0answers
36 views

Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
3
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27 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
3
votes
0answers
121 views

Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$. Is ...
3
votes
0answers
85 views

Decomposition of disjoint cycles

Work out the decomposition in disjoint cycles for the following. a) (14)(12345) = (15)(234) b) (12)(2345) = (12345) c) (12)(23)(34) = (14)(24) d) (13)(1234)(13) = (143)(2) Can anyone tell me ...
3
votes
0answers
168 views

Circular permutations - $n$ sitting at a round table without repeating neighbors

I hope this isn't a duplicate - the problem is to find the number of ways of sitting $n$ people (who initially were sitting in the order $1, 2, \dots,n$, with $1$ and $n$ being neighbors) at a round ...
3
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0answers
66 views

Probability/selection problem

Assume that we have $N$ items of $M$ distinct types in a closed bag. We also have $K$ bowls $(K \leq M)$ that can hold only items of same type. In the beginning bowls are empty. And bowls can hold a ...
3
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0answers
56 views

Dinner group rotation. Sixteen couples. Four couples per house. Each couple to meet all the others, no repetition.

I want to set up a rotation of sixteen couples with four couples per house so that all couples eventually have dinner together, no repetition. Each couple is to host one dinner. Meetings are monthly ...
3
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0answers
83 views

Counting number of distinct systems

This is an enumeration problem in conjonction with some lottery problems. Given an integer $N \ge 5$. Let a ticket be a set of 5 distinct integers between $1$ and $N$. Given an integer $T$ between ...
3
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0answers
120 views

Is there a simple way to show that wreath product is associative?

Is there a simple way to show that wreath product is associative? If your proof is short, please write it explicitly. Thank you.
3
votes
0answers
203 views

Combinatorics Issue without repetitive combinations

We have 26 Boxes Labeled: Box 1, Box 2, Box 3 and so on. The boxes are in a specific order. We also have 15 rocks. Rocks are all identical. meaning Rock 1 is no different then Rock 2, or does not have ...
3
votes
0answers
129 views

Defining a signed involution

We define the displacement of $\pi$ as $\mathrm{disp}(\pi)=\sum_{i=1}^n|\pi(i)-i|$. I know that it's even. Could you help me to find a good signed involution of the set of permutation with ...
3
votes
0answers
558 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
2
votes
0answers
34 views

give a group that is isomorphic to the figure.

I think if I get help with one of these I should be good on the rest. 23) is concerned with figure a) It has one symmetry and 4 possible points. seems like it would have two elements that map to ...
2
votes
0answers
21 views

How to combine possible permutations of two sets to find number of combined permutations

I hope the title accurately describes the question. I have a question that asks: There are 7 male swimmers and 5 female swimmers. If there is a gold, silver, and bronze medalist male swimmer, and a ...
2
votes
0answers
8 views

Is there a name for these oscillations in the self-similarity of a set under the action of a cyclic group?

I don't know much about group theory and card-shuffling theory, so this may already have a name I don't know about. I often shuffle a deck of cards using a method that is defined by a particular ...
2
votes
0answers
19 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
2
votes
0answers
46 views

Number of permutations of order m

Is there a closed form for the number of permutations (on n letters) that have order m? If not, is there a tight upper bound?
2
votes
0answers
52 views

Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

In class they gave us a problem, After spending a long time trying to solve it, I turn to you =] Let $H_1,H_2,H_3, K_1,K_2,K_3 \le G$ be simple groups, $G=\{{h_1}{h_2}{h_3}:h_1\in H_1,h_2\in ...
2
votes
0answers
78 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
2
votes
0answers
37 views

Conjugacy class A(4)

I want to find all conjugacy classes of $A(4)$. So basically what I did, I took all elements of $A(4)$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, ...
2
votes
0answers
41 views

Is my solution correct? (If $S_\Omega \cong S_\Delta$ then $|\Omega|=|\Delta|$)

I've tried to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 151, Exercise 8). Here $\Omega$ is an infinite set, $D \leq S_\Omega$ is the subgroup of permutations ...
2
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0answers
30 views

Need a hint with permutations and pigeonhole-principle question

let $\pi_1,\pi_2,\pi_3\in S_{28}$. Help me prove that there are two sub-sequences of 28 with length 4 $i_1< i_2 <i_3<i_4,\ and\ \ j_1<j_2<j_3<j_4$ so that $\pi_q(i_n)=\pi_p(j_n)$ ...
2
votes
0answers
55 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
2
votes
0answers
19 views

Different permutations of n identical groups with sizes a, b, and c.

My question is trying to solve how many paths there are from $(x,y)$ to $(tx, ty)$ 3 possible moves are allowed at each step: increment $x$ by $1$ increment $x$ by $2$ increment $y$ by $1$ I know ...