0
votes
0answers
14 views

Collecting terms of a hard linear equation

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
1
vote
1answer
21 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
19 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
60 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
75 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
67 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
72 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
34 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
51 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
42 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
120 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
35 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
42 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
61 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
12 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
30 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
35 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
39 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
82 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
83 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
75 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
168 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
102 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
44 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
45 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
51 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
39 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 ...
2
votes
1answer
62 views

Number of orbits of $M_n(K)$ under the action of $\mathbb{GL}_n(K)$

It's easy to prove that given two nxn matrices X,Y with coefficient in a field K,with same rank, there are $A,B \in \mathbb{GL}_n({K})$ such that $AXB=Y$. But clearly it's not true if we just search A ...
2
votes
3answers
39 views

Algebra question / conversion of ranges

Greets All Forgive me if I'm using the wrong terms but I'm trying to sync up two number ranges together. Example: I have two x axis (ranges) I would like to equate with each other ...
15
votes
2answers
340 views

A “What's my vector?” game

Alice chooses a binary vector $V$ of length $n$ which is unknown to Bob. In each round Bob can choose any number of indices $i$ and then Alice tells Bob how many of the $V_i$ are set to $1$. The ...
1
vote
0answers
18 views

Show C is not 1-error correcting by using Slepian decoding

Let C $\subseteq$ $ \mathbb{Z}_2^5$be a linear code with generator matrix $$G=\begin{bmatrix}1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1\\0 & 0 & 1 & 0 & ...
2
votes
5answers
339 views

How many numbers must be selected from the set

How many numbers must be selected from the set {1, 3, 5, 7, 9, 11, 13, 15 } to guarantee that at least one pair of these numbers add up to 16, explain your answer?
0
votes
1answer
40 views

Combinatorics and Linear Alegbra

$$T = \{1011, 0112, 2101\} \subset Z_3^4$$ Is there any efficient way to find the span for set T other than checking all 27 possibility? If so, how to do it?
1
vote
2answers
89 views

Calculate the determinant of the matrix.

Calculate the determinant. \begin{bmatrix} C_{n}^{p+n} & C_{n}^{p+n+1} & \dots & C_{n}^{p+2n} \\ C_{n}^{p+n+1} & C_{n}^{p+n+2} & \dots & C_{n}^{p+2n+1} \\ \vdots & ...
2
votes
1answer
62 views

Using matrix theory to solve this problem

I'm sorry that I couldn't find a better title for this. I was wondering if my solution is valid for the following problem, or if I've made some mistake. Problem: Let $N=\{a_1, \dots, a_n\}$ be a ...
2
votes
1answer
78 views

How many Jordan normal forms are there when the characteristic polynomial is $(\lambda+4)^5(\lambda-2)^2$?

Let $A\in M_7(\mathbb{C})$ be a matrix in with the characteristic polynomial $p(A)=(\lambda+4)^5(\lambda-2)^2$. I need to find all Jordan normal forms for this. I think that i can use that the ...
2
votes
1answer
92 views

Linear algebra and combinatorics. For a family with even size sets and even intersections prove that $|F| \le 2^{n/2}$

Let $F \subset P(n)$ be a family such that for all i and j $ |f_i \cap f_j|$ and $|f_i|$ are even Prove that $|F| \le 2^{n/2}$ Now I think we go by contradiction and say if $|F| \ge 2^{n/2}+1$ ...
4
votes
0answers
104 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
1
vote
1answer
73 views

Proof for existence of exactly one solution for the number of marbles in each box

There are four boxes A, B, C and D containing marbles. Two boxes are randomly selected and the number of marbles in each box is summarized. This procedure is repeated five times with the ...
1
vote
0answers
24 views

Efficient evaluation of the inverse of a triangular matrix on a vector

I have this matrix that interests me. It arises when we try to express the norm of a $(p,p)$-form on an $n$-dimensional vector space in terms of (squares of) traces of the form with respect to the ...
1
vote
1answer
29 views

linear dependncy of a random vector with respect to a reduced row echelon form in a finite field

Given a matrix with elements from a finite field $\mathbb{F}_q$, $A\in\mathbb{F}_q^{N\times M}$, where $q$ is the size of the field, $N<M$. Suppose that $A$ in the reduced row echelon form. ...
1
vote
0answers
38 views

convexity in oriented matroid theory

I would like to try to solve the following problems. If someone knows how to prove at least part (a), could you show me the proof? I am having a LOT of trouble understanding oriented matroid theory. ...
1
vote
0answers
85 views

Number of ways to decompose the space $\mathbb F^n_2$ into a direct sum of two spaces

How many ways can $\mathbb F^n_2$ be decomposed into a direct sum of two subspaces? Basically how do I find the number of decompositions $\mathbb F^n_2 = \mathbb F^k_2 \bigoplus \mathbb F^{n-k}_2$ ...
0
votes
1answer
19 views

Is there any weighted graph which smallest eigenvalue of its adjacency matrix is greater than 1?

Is there any weighted or unweighted graph which smallest eigenvalue of its adjacency matrix is greater than 1?
2
votes
0answers
79 views

Does a matrix represent a bijection

We have a square binary matrix that represents a connection from rows to columns. Is there a way to tell if a bijection exists (other than checking for all possible bijections and iterating through ...
1
vote
1answer
49 views

Normalizing a matrix with row and column swapping

How do you canonicalize a matrix over column- and row-swap operations? Or more specifically, does there exist a function f(M) such that ...
0
votes
0answers
30 views

Graph Theory and sandpiles

Using Matrix-tree theorem how could we conclude the order of S(G) is the sum of the weights of G's directed spanning trees into s where S(G) is the sandpile group of a sandpile graph G=(E,V,s).
0
votes
1answer
519 views

Finding the Greatest Coefficient in a Binomial Expansion?

when I do this question, I try not using the: $(n-k+1)/k * b/a$ formula, but rather the $T(k+1)/T(k) ≥ 1$ formula. However, when I do it like that, I get the wrong answer - which is probably a simple ...