4
votes
0answers
42 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
2
votes
2answers
39 views

The Number of Increasing Vectors $(x_1,…,x_k)$ Satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <…<x_k$

I want to find the number of increasing vectors $(x_1,...,x_k)$ satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <...<x_k$. Examples of vectors satisfying these conditions Let $n =5$ and ...
0
votes
1answer
52 views

Is there a binary [10,6,4] code?

Using the sphere padding packing bound formula I can conclude that 1 + 12 + 66 $\ge$ $2^{6}$ which indicates that there MAY be a binary [10,6,4] code, however I cannot prove that there is. How can I ...
1
vote
1answer
42 views

Finding maximum no of $1$'s

We are given a matrix $A \in M_n (F)$ such that all its entires are either $1$ or $0$. I need to find the maximum number of $1$'s that can be in matrix $A$ so that it is still invertible. My try : ...
0
votes
0answers
18 views

Investment Strategies(Integer Solutions Of Equations)

You have \$$20$k that must be invested among $4$ possible opportunities. Each investment must be in multiples of \$$1$k, and there are minimal investments that need to be made if one is to invest in ...
2
votes
1answer
42 views

Algebraic Combinatorics about a Finite Graph

Here is a problem listed on a book 'Algebraic Combinatorics' by Richard P.Stanley. Let $G$ be a finite graph with at least two vertices. Suppose that for some $l \ge 1$, the number of walks of ...
0
votes
0answers
20 views

Some theorem of Newton (discrete Taylor expansion)

Let $\Delta$ be the forward difference operator, $\Delta f(x)=f(x+1)-f(x)$. Is there an elegant way to prove that for every $f\in\mathbf{Q}[x]$ (of degree $n$, say) the equality $$f(x)=\sum_{k=0}^n ...
0
votes
1answer
35 views

How to prove the equivalence of 2 affine spaces given that one is the subset of the other one?

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
2
votes
1answer
44 views

Number of integral soultions to linear equations without unit coefficients

To determine the number of integral solutions for the linear equation $$ x_1+x_2+x_3+\cdots+x_k = N$$ we have an expression $$ ^{N+k-1}C_{k-1}$$ But I want to know if the coefficients of ...
0
votes
0answers
42 views

Proving that number of codes with even weight is the same as number of codes with odd weight for a specific code book

Consider the $[n,n]$ code-book $C_0=\{0,1\}^n$ with $n$ being odd and the codes $c_i \in C_0=[c_1,c_2,...,c_{2^n}]$ being sorted in the ascending order of hamming weight (from $0$ to $n$). Now let's ...
2
votes
0answers
45 views

Density of Pythagorean triples

We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$. ...
1
vote
2answers
16 views

Upper bound for the number of maximally linearly independent subsets of a set

Let $A \subseteq \mathbb R^n$, with $|A| = m > 0$ (finite, we can assume $2^n > m > n$). I want to upper bound the number of bases of $\mathbb R^n$ contained in $A$. I do realize that a very ...
1
vote
1answer
67 views

Odd town Even town explanation.

I am struggling to understand the solution to the following problem: If $\mathcal F\subset 2^{[n]}$ such that for each $F_1$ and $F_2$ in $\mathcal F$ we have $|F_1|,|F_2|\equiv 1 \bmod 2$ and ...
1
vote
1answer
49 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
3
votes
1answer
33 views

Number of possible rectangles from at most N identical squares

I was looking to find the number of distinct rectangles possible from at most $N$ identical squares. (Two rectangles are distinct if one cannot be rotated to obtain another) e.g for $N = 6$ , $8$ ...
0
votes
1answer
39 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...
1
vote
1answer
28 views

permutations with a given condition!

What will be the number of permutations of n different things, taken r at a time,when p particular things is to be always included in each arrangement? I know the answer to this question but could not ...
1
vote
1answer
39 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
0
votes
1answer
27 views

Linear Constraints Solution Existence

how can one decide if $$A*t\ge b$$ $A$ is a Matrix with integer Entries and $t$ is a Vector with integer Entries, $b$ is a fixed Vector with integer Entries exists?
0
votes
1answer
29 views

Finite sum equaling Kronecker Delta

could anyone help understand how $$\sum_{j=0}^{n-r}\binom{n-r}{j}*(-1)^{j} = [1 + (-1)]^{n-r}$$ I see that if $j=0$, i get $1=1^{n-r}$, and if $j=n-r$, i get $(-1)^{n-r},$ but what about the rest of ...
-2
votes
2answers
29 views

common multiple polynominal time

Given $n$ rational numbers. Is there a polynominal time algorithm to compute a common denominator? My idea was for each number search for $k_i$ so that $k_i \cdot n_i$ is integer. Then the solution ...
0
votes
0answers
59 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
0
votes
0answers
37 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
3
votes
2answers
31 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
6
votes
1answer
141 views

My fun conjecture about linearly independence

In the $\mathbb{R}^n$ vector space, there are distinct $m$ vectors $v_i$'s ($1< i\leq m)$ such that each component has value 0 or 1. Let $A_i$ be the set of $j$'s where $j$-th component of $v_i$ ...
2
votes
0answers
48 views

solving recurrence relations for functions with more than one variable

Is there a way to find formula for a function on more than one variable which is given by recurrence relation with some initial conditions? e.g.if one knows the value of f(n,p,l) for all p,l where ...
0
votes
3answers
37 views

How many are possibilities to build count $n$ summing $k$ other counts?

I have got an integer $n$. I have to build it by summing $k$, not necessary, different integers. Is there any overall formula to count how many are possibilities to build count $n$ summing $k$ other ...
2
votes
2answers
47 views

Sum of $\mathbf{x}\mathbf{x}'\mathbf{x}\mathbf{x}'$ when $\mathbf{x}$ is a binary vector of length $n$

Let's say $\mathbf{x}$ is an $n\times 1$ column vector where each entry can be either 1 or -1. There can be $2^n$ possibilities for this vector. Let $\mathbf{x}'$ be a transpose of $\mathbf{x}$. I ...
0
votes
0answers
20 views

Name search for special Linear Mixed Integer Programm

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
0
votes
1answer
26 views

How many codimension 1 submodules of $(\mathbb{Z}/2)^n$ are there?

Let $\mathbb{Z}/2$ denote the ring of two elements. How many codimension-1 submodules of the $n$-dimensional free module $(\mathbb{Z}/2)^n$ are there?
3
votes
0answers
78 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
20
votes
2answers
458 views

Which vectors can give zero inner products forever

For even positive integer $n$, consider an $n$-dimensional vector $v$ such that $v \in \{-1,0,1\}^n$. Now consider an infinite dimensional vector $w$ with $w_i \in \{-1,1\}$ and define $I_k = ...
2
votes
0answers
31 views

Number of elements and number of different basis of $\mathbb F_5^3$

Let $\mathbb F:=\mathbb F_5$ the field with five elements. (i) How many elements has $\mathbb F^3$? (ii) How many different basis has $\mathbb F^3$? My idea: (i) $\mathbb F^3$ has $5^3$ elements. ...
1
vote
1answer
90 views

Question About the Solutions to the Eight Queens Problem [closed]

How is $a_{15}n_8e_9k_5f_{10}d_7b_4m_6$ a solution to the Eight Queens problem? J. W. L. Glaisher, On the Problem of the Eight Queens, Philosophical Magazine, 1847 says that each one of these terms ...
-1
votes
1answer
37 views

abandon a column, also $n$ different row vectors

$A$ is a $n\times n$ matrix, whose $n$ row vectors are all different. then, we can get rid of one column of $A$(there exist a column, we abandon this column ), such that the new $n\times (n-1)$ ...
2
votes
0answers
45 views

diagonal action induces permutation

Suppose one has two $n$-tuples of complex numbers $(c_1,\dots,c_n)$ and $(z_1,\dots,z_n)$ such that all $c_i$, $z_i$ are nonzero, and $$ (c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)}) $$ ...
-1
votes
1answer
55 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
1
vote
1answer
30 views

(Counting problem) very interesting Modular N algebraic eqs - for combinatorics experts

We have some attempt to numerically solve this math problem, which means that we like to count the number of independent solutions of this set of six of modular N algebraic equations: $$ (1) x_1 ...
3
votes
0answers
25 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) \\ ...
2
votes
2answers
80 views

(Counting problem) very interesting Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ (1) \quad x_1 y_2 \equiv x_2 y_1 \pmod{N}\\ (2) \quad x_1 y_3 ...
2
votes
0answers
99 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
0
votes
1answer
87 views

List all the permutations of {1,2,3,4}. Which are even, and which are odd?

The answer is: There are 24 permutations. The 12 even permutations are: id , (1 2 3 4) , (1 3 2 4) , (1 4 2 3) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3). The ...
2
votes
2answers
119 views

Powers of permutation matrices.

Let $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time). How to show that $$P^{\ n+1} = I$$? Is it true that, ...
0
votes
0answers
39 views

Counting problem about sub-matrices

EDIT (along the lines suggested by @sea turtles): Given a prime power $q$ and a positive integer $x\gt q$, how many subsets $A\subseteq{\bf F}_2^q$ have size $x$ and contain a subset $B\subseteq A$ ...
1
vote
3answers
51 views

Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$ [duplicate]

The question: Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$. So far I have $S_n = \sum_{k=1}^{n} H_k = \sum_{k=1}^{n} ...
1
vote
0answers
61 views

Prove that the minimum of row sums of a nonnegative symmetric matrix is preserved

Let $A$ be an $n\times n$ adjacency (nonnegative, irreducible and symmetric) matrix with zeros on the diagonal. Denote $i$-th row sum of $A^k$ as $r^{(k)}_i$, where $k\geq1$. I want to prove that if ...
0
votes
1answer
43 views

psittacism: Fundamental Theory of Time

This question is in reference to the programming question found here. What method of approach should I be thinking of if I have a list of lectures A, B, and C, and discussions D, E, and F, that are ...
1
vote
4answers
165 views

How to tell if two matrices are equal up to a permutation

Given two real rectangular matrices A, B how can I tell if they are equal up to a permutation of their rows/column without trying all possible permutations? (This is closely related to the question I ...
0
votes
0answers
38 views

Summing a particular product of binomial coefficients

I expect this is elementary, but I can't find a closed form. Let $a_i$, $i=1,...,m$, be a sequence of natural numbers and $n>\sum a_i$. What is the value of the sum: ...
1
vote
0answers
45 views

What type of formula am I looking for?

Let say you have a list of items with 3 columns, two are statistical the third is just a name. The statistical categories you have are Points, and Salary. You have 10 different options. Each Row ...