Tagged Questions
2
votes
0answers
36 views
Testing combinatorial species for isomorphism
Given a system of species equations that specifies two species, is there an algorithm to test if they are isomorphic?
Testing for isomorphism can be done by testing the equality of the coefficients ...
9
votes
1answer
188 views
What can we say about the size of $HK\cap KH$ when $HK\neq KH$?
If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is
If ...
2
votes
2answers
56 views
Why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
0
votes
0answers
49 views
Manual Solutions [closed]
Somebody know any manual of solutions of these books:
1) Measure and Integral -- An Introduction to Real Analysis, Richard L. Wheeden & Antoni Zygmund;
2) Introduction to Topological Manifolds, ...
2
votes
1answer
44 views
Confusion related to a derivation
I have a confusion related to a derviation
If k is any natural number and k' is largest natural number that is strictly smaller than 7/8k. Then how come
$7/8k - k' >= 1/8$
4
votes
0answers
25 views
Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code
I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection.
Let $C$ be some extended ...
4
votes
1answer
72 views
Counting binary operations on a set with $n$ elements
I am trying to solve following problem but not able to find any way to proceed.
Let $S$ be a set having $n$ elements. Can we count about number of binary operations that can be defined on a set? Can ...
3
votes
1answer
41 views
How many unique Hamming (7,4,3) Codes are there?
I cannot find an answer to this on Google so I thought I would ask here. By unique I mean "distinct sets of codewords."
By my count, there are $7!$ ways to choose an ordering of message bits and ...
7
votes
1answer
88 views
How do combinations (not permutations) relate to group theory?
First question. I'm just generally curious about combinations in group theory. How do they relate?
If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
6
votes
1answer
97 views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
"... is the ...
4
votes
1answer
85 views
A Question on the Young Lattice and Young Tableaux
Let:
$\lambda \vdash n$ be a partition of $n$
$f^\lambda$ - number of standard Young Tableaux of shape $\lambda$
$\succ$ - be the covering in the Young Lattice (that is, $\mu \succ \lambda$ iff ...
2
votes
1answer
42 views
Standard Young Tableaux and Bijection to saturated chains in Young Lattice
I'm reading Sagan's book The Symmetric Group and am quite confused.
I was under the assumption that any tableau with entries weakly incresing along a row and strictly increasing down a column would ...
5
votes
1answer
113 views
Why is $S_5$ generated by any combination of a transposition and a 5-cycle?
Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
2
votes
3answers
41 views
Computing $\langle (13746) \rangle$ in $S_7$.
How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
1
vote
2answers
61 views
Prove an inequality in a group ring
Let $$G=\bigoplus_{n\in\mathbb{Z}}\left(\mathbb{Z}/2\mathbb{Z}\right)_n$$ be a group, and for any $n\in \mathbb{Z}$, denote $\delta_n$ to be the element in $G$ with $n$-th coordinate $1$ and zero at ...
1
vote
2answers
201 views
Combinatorics in finite cyclic groups
Discuss the following.
I got a good platform to remove all me quarries from my mind by positing the problems like this. Thanks again for support.
1) Find the minimum elements must be selected from ...
6
votes
2answers
90 views
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
Assume that $\pi$ is an isomorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. Anyone ...
6
votes
0answers
68 views
Information-theoretic aspects of mathematical systems?
It occured to me that when you perform division in some algebraic system, such as $\frac a b = c$ in $\mathbb R$, the division itself represents a relation of sorts between $a$ and $b$, and once you ...
3
votes
0answers
87 views
$\sum _{i=1}^{n} \sum _{j=1}^{n} \sum _{k=1}^{n}\sum _{l=1}^{n} A(i,j)A(i,k)A(i,l)A(j,k)A(j,l)A(k,l) $
I want to find an efficient algorithm for calculating a sum of products with entangled indices. For example, $\sum _{i=1}^{n} \sum _{j=1}^{n} \sum_{k=1}^{n} A(i,j)A(j,k)A(k,i)$, where A(i,j) is the a ...
16
votes
3answers
490 views
What's next for me?
I'm in my last year of undergrad, and I would like to do original research for my senior thesis. I am already published in finite group theory and am looking for a new topic to study.
I have taken ...
1
vote
1answer
136 views
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime.
I'd like to start off by acknowledging that I know there are many posts relating to similar ...
0
votes
1answer
111 views
Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).
I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics).
I'm not interested in the usual sense dictionary ...
5
votes
1answer
149 views
Mutual set of representatives for left and right cosets: what about infinite groups?
Let $G$ be a group and $H$ a subgroup of $G$. If $G$ is finite, then according to Philip Hall's "marriage theorem" there is a left transversal $T$ of $H$ in $G$ (that is, $T$ contains precisely one ...
48
votes
2answers
4k views
More than 99% of groups of order less than 2000 are of order 1024?
In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.
Is this for real? How can one deduce this result? ...
30
votes
1answer
1k views
Is War necessarily finite?
War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules.
A ...
3
votes
1answer
51 views
Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.
As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the ...
2
votes
1answer
58 views
Commutants for matrices
Let ${\mathbb K}={\mathbb R}$ or $\mathbb C$. Let $V$ be a vector space over $\mathbb K$ and fix a basis $\cal B$ of $V$. We say that a family of vectors of $V$ is nice (relatively to $\cal B$) if
if ...
0
votes
0answers
32 views
Normed Division Algebras and Combinatorial Hierarchy
Is there a mathematical identity between the 4 normed division algebras and the 4 levels of the combinatorial hierarchy?
0
votes
3answers
54 views
Subgroups of $\Bbb Z_n$
Consider the cyclic group $\Bbb Z_n=\{1,2,\cdots n\}$ under addition modulo $n$ and for some non zero $a\in \{1,2,\cdots n-1\}$ let $\langle a\rangle=H\le \Bbb Z_n$ of order $q$. I wish to show that ...
5
votes
3answers
77 views
$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?
I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
1
vote
1answer
60 views
Abstract Algebra questions related to Burnside's Lemma
I am trying to understand these two derivations in class.
Here $g \in G$ and $x \in S$.
$\displaystyle\sum_{x \in S} |G_x| = \sum_{\text{orbits} \hspace{ 1mm} Gx} \hspace{ 1mm} \sum_{y \in Gx} |G_y| ...
4
votes
1answer
183 views
Graphs with a unique $3$-path free acyclic orientation up to isomorphism.
Let $\Gamma$ be a simple, $3$-colorable graph such that, up to isomorphism, there exists exactly one acyclic orientation of $\Gamma$ that does not contain a directed 3-path. (To be clear, when I say ...
5
votes
1answer
97 views
Number of specific partitions of a given set
Let U be the set $U=\{(1,2,3,\ldots,2^m)\}$. Let $A$ and $B$ partitions of $U$, such that $A \cup B$ is the set $U$, and their intersection is empty, and adding the elements of the first set is the ...
2
votes
1answer
87 views
Polynomials and partitions
There is a question I have based on the fact:
If you take a quadratic polynomial with integer coefficients, and take the set (1,2,3,4,5,6,7,8), and make a partition A=(1,4,6,7), and B=(2,3,5,8), and ...
1
vote
1answer
109 views
How many $1$'s could there be in this sequence? Matrix, operator?
For each $(i,j)\in \mathbb{N}^2$, $a(i,j)=1$ or $0$, and 1) $a(i,i)=0$ for all $i$; 2)for fixed $i$, there is at most one $j$ such that $a(i,j)=1$. Suppose we know that there is a finite $\kappa$ such ...
8
votes
1answer
145 views
Permutations: Given $P^4$, how many $P^1$s are possible?
Let $P^0$ be the identity tuple $(1,2,...,N)$
Let $P^{i+1}$ be the tuple after a permutation $P$ is applied to $P^i$.
For example, if $P$ is $(2,1,3,6,4,5)$ than:
$$\begin{align}
P^0 &= ...
2
votes
1answer
139 views
Group structure from involutions, exercise devised by Richard Brauer.
The following is an idea originally communicated by Richard Brauer. I'm having difficulty following some of the combinatorial elements.
Let $G$ be a finite group containing exactly two conjugacy ...
2
votes
0answers
53 views
Algebra involved in computations with (or extensions of) a probability measure on a lattice
Suppose we have a probability measure $\gamma$ defined on the $d$-dimensional lattice $\mathbb{N}^d$. Let us say $d = 5$ for simplicity. I will also write $\gamma_{1:5}$ for this measure and write say ...
5
votes
2answers
657 views
Edge coloring of the cube
We have a cube and we are coloring its edges. There are three colors available. We say that the two colorings are the same if one can obtain a second by turning cube and permuting colors. Find the ...
2
votes
0answers
85 views
Properties and extensions of the $n!$ formula for $e^{-1}$? [closed]
So I recently re-encountered the following limit: $$\lim_{n\rightarrow \infty} \dfrac{(n!)^{1/n}}{n}=\dfrac{1}{e}$$
I began to wonder about a few things from this relation:
(i) I notice that ...
2
votes
1answer
60 views
Minimal subset of $x_1, x_2, …, x_{100}$ that XORs to $y$
Given a 64-bit positive integer $y$ and a set of $100$ $64$-bit positive integers: $X = \{ x_1, x_2, \dots, x_{100} \}$
I want to find a smallest possible $Z = \{z_1, z_2, \dots, z_n\} \subset X$, ...
4
votes
1answer
94 views
Is $(p,\epsilon,p)$ a path of an automaton?
$A$ is an alphabet. An automaton over $A$ can be defined
as a set $A_0 = (Q, E, I, T),$ where $Q$ is the set
of states, $E \subseteq Q \times A \times Q$ is the
set of edges or transition, $I, T ...
1
vote
1answer
67 views
Does a binoid contain an empty word?
$A$ is a finite alphabet. $A^*$ is the set of finite words or the free monoid generated by A.
$A^w$ is the set of infinite words generated by A. Denote $A^\infty=A^*\bigcup A^w$.
$X$ is a set of ...
2
votes
0answers
84 views
What is the antipode in a Combinatorial Hopf Algebra (or graded bialgebra)?
In several papers I've seen on Combinatorial Hopf Algebras, the algebra and coalgebra structures are described, but no antipode is defined. CHAs generally have a natural grading, and are of finite ...
1
vote
1answer
47 views
Similar to Macmahon's theorem, why does $\sum t^{\text{maj}(w)}=\sum t^{\text{inv}(w)}$?
In 1913 Percy MacMahon proved that the distribution of the major index on all permutations of a fixed length is the same as the distributin of inversions. I'm trying to understand the identity
$$
...
3
votes
2answers
174 views
Identity of signed permutations?
Let $B_n$ denote the group of signed permutations on $n$ letters. Is there a good explanation or understandable way to see why
$$
\sum_{w\in B_n}q^{\text{inv}(w)}=(2n)_q(2n-2)_q\cdots(2)_q?
$$
I've ...
2
votes
2answers
256 views
Product of all irreducibles with degree divisible by $n$ in $\mathbb{F}_{q^n}$?
In the finite field of $q^n$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to simply be $X^{q^n}-X$. Why is this?
I understand that $q^n=\sum_{d\mid ...
3
votes
2answers
140 views
Unusual version of the binomial theorem?
This was an old problem I had years ago, but never really solved. Maybe it can be cracked here?
The situation is as follows.
Denote by $\mathbb{Q}(q)[X,Y]$ the algebra of polynomials over ...
4
votes
1answer
377 views
On the group of signed permutations?
Let $B_n$ be the group of signed permutations, which is a Coxeter group acting on $\mathbb{R}^n$ with Coxeter generators $\sigma_i=(i\; i+1)\in S_n$ and the change of sign ...
2
votes
1answer
61 views
Identity of generating function of weights of multiset cycles.
A few days ago I asked this question on a generating function of multiset cycles.
There is was shown that $\prod_C(1-w(C))^{-1}=\sum_{\pi}w(\pi)$ where $w(\pi)$ is the weight a a multiset permutation ...
