0
votes
1answer
28 views

What is the 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 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 guess ...
1
vote
1answer
26 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
35 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
26 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
26 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 ...
-1
votes
2answers
27 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
57 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
34 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
29 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
140 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
43 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
36 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
46 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
18 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
22 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
76 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
457 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
81 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
44 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
28 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
24 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
78 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
97 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
82 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
100 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
38 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
46 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
55 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
143 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 ...
2
votes
2answers
71 views

Stirling Number of First Kind

How i can calculate stirling number of first kind $s(n,k)$. I need to calculate it for $n$ up to $100$. I need to calculate the $s(n,k)$ modulo $x$. Here $x$ is a finite integer.
0
votes
0answers
31 views

Number of configurations? [duplicate]

I have an array of n elements with all intialised to zero I also have M queries. Each query has a starting and ending index. In each query i just changes the array elements belonging in that range. If ...
0
votes
0answers
14 views

Is there a general formula for counting number of canonical forms given the minimal polynomial?

Suppose you have an operator $T$ on a vector space $F^n$, and you're given the minimal polynomial $m_T(x)=p_1(x)^{a_1}\cdots p_k(x)^{a_k}$, where $\sum a_i=\deg(m_T(x))=d\leq n$. Is there a general ...
0
votes
0answers
35 views

Can one find arbitrarily large subsets of a vector space of dimension n, such that any subset of n vectors is a basis?

I thought of this mildly interesting question earlier this evening: Given a vector space $V$ of dimension $n$, for what values of $m > n$ is it possible to create a set $S$ of $m$ vectors such ...
0
votes
1answer
41 views

Combinatorial exercise

A group of 15 people go visit a city with 150 bar. At the end of the day one of those bar contains 8 people, the another one contains the other 7 people. How many ways can we get this situation? ...
0
votes
0answers
43 views

Producing integer combinations of irrational numbers in sequence?

Let $\mathbf{w}=\{w_0,w_1,\cdots,w_n\}$, $\mathbf{k}_i=\{k_0^i,k_1^i,\cdots,k_n^i\}$ and $\mathbf{m}_i=\{m_0^i,m_1^i,\cdots,m_n^i\}$, where $w_j\in\mathbb{R}$, $k_j^i\in\mathbb{Z}$ and ...
1
vote
1answer
84 views

Proving summation identities [duplicate]

How would one go about proving the following identities? $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i}{z_i-z_j} = \frac{n(n-1)}{2}$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^2}{z_i-z_j} = ...
0
votes
2answers
102 views

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly m =< n. In general, for computational purposes, what I have managed to do ...
0
votes
1answer
90 views

How many moves (shifts) are needed to sort an unsorted sequence of numbers $1$ to $n$ in ascending order?

I have the LUP decomposition of a matrix. The determinant can be found from the formula: $$\det(A) = \det(P^{-1}) \det(L) \det(U) = (-1)^s \left( \prod_{i=1}^n l_{ii} \right) \left( \prod_{i=1}^n ...
3
votes
4answers
187 views

Coefficient of $x^n$ in the series

How will we find the coefficient of $x^n$ in the following series: $$(1+x+2x^2+3x^3+...)^n$$ Please suggest if there is some formula or if it can be computed using the computer in $\log n$ time. I ...
0
votes
3answers
106 views

Proof involving a summation

How would I go about proving that $\sum_{i<j} 1 $ = $ n\choose 2$ and $$ \sum_{i<j}(x_i +x_j) = (n-1)\sum x_i $$ I understand the intuition behind the statements. I'm just unsure of how to ...
5
votes
1answer
69 views

Looping over $k$-element subsets by switching elements

I would like to iterate over the $k$-element subsets of $\{1,2, \dots, n\}$ in a natural way by switching elements. Two subsets $v,w$ are adjacent if $|v \cap w| = k-1$ or equivalently if their ...
3
votes
1answer
49 views

On a partition of non-zero vectors into some subspaces

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $m \geq 2$ and $V_1$, ..., $V_m$ are non-zero subspaces of $V$ such that every non-zero vector belongs to one and only ...
3
votes
0answers
52 views

Deducing that polynomials span

Let us say that we are dealing with a countable family of polynomials with real coefficients in $n$ indeterminates that commute. Are there any known/common nice systematic ways to tell if their span ...
0
votes
1answer
46 views

Designing an algorithm to determine if a linear combination of k-1 sets is contained in the k-th set .

I am trying to solve the following problem - given $k$ sets : $A_1,A_2,...,A_k$ containing $O(n)$ integers each I need to design an algorithm that will determine if there is such a group of elements ...