# Tagged Questions

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### the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
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### Find solutions to given equation

Find all integer solutions $x$ for $0 < x < 10^9$ of the equation: $$x=b\cdot s(x)^a+c,$$ where $a$, $b$, $c$ are some predetermined constant values and function $s(x)$ determines the sum of ...
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### Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
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### Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
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### Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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### Direct combinatorial proof of a sum identity on formal Lagrange polynomials

Let $k$ be a field and $K=k(x_0,x_1,\ldots, x_n)[x]$. Define $$\mathcal{L}_k(x)\triangleq \prod_{\substack{j=0\\ j\ne k}}^n\frac{x-x_j}{x_k-x_j}.$$ Is there a purely combinatorial way to show ...
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### Find all divisors of an polynomial (simple combinatorics)

I want to implement the multivariate Kronecker factorization algorithm and at one stage I need to find out all divisors of a polynomial $u(y,\dots)$. I already know the irreducible factorization of ...
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### stuck trying to solve nonhomogeneous recurrence relation

hello and happy new year! I'm trying to solve this question: We are required to find the solution (direct formula) of the following recurrence relation: $b(n)=b(n-1)+n-1$, $b(0)=0$. What I did: I ...
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### How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
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### Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
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### Defining irreducible polynomials recursively: how far can we go?

Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such ...
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### Sum of multinomial coefficients with constraints

The title doesn't reflect the question properly, since I don't know enough about combinatorics to get it right, here. Feel free to change the title. From the multinomial theorem, we can deduce, that ...
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### $x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$

Show that the number of solutions in nonnegative integers of the inequality $$x_1+x_2+\cdots+x_n\leq M,$$ where $M$ is a nonnegative integer, is $C(M+n, n)$.
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### Throw a die three times, and get maximum number of different sums.

The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest ...
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### dimension of space of polynomials

Let $\mathcal P_k^n$ be the space of all polynomials of degree $\leq k$ in $n$ variables. Prove $\dim\mathcal P_k^n = {n+k\choose k}$. I tried showing this by taking $n\in\mathbb N$ an arbitrary ...
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### Newton's binomial problem

It is known that in the development of $(x+y)^n$ there is a term of the form $1330x^{n-3}y^3$ and a term of the form $5985x^{n-4}y^4$. Calculate $n$. So, I know that the binomial formula of Newton ...
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### Writing a sum as a fraction

Express $$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible. Hmm. How to do it? Wolfram is really stupid on ...
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### express Pochhammer symbol $(x)_n$ as a polynomial of order $n$ in $x$

Define $$(x)_{n}=x(x-1)(x-2)...(x-n+1)=\prod_{k=1}^{n} (x-k+1)=\sum_{k=0}^n a_k x^k$$ Q: what is the closed-form expression for $a_k$ ?
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### Number of coefficients of multivariable polynomial

Let $g \in \mathbb{F}[x_1, \dots, x_n]$ be a polynomial of degree $d$ with $n$ variables. Number of its coefficients is ${n+d \choose d}$ Is there an easy proof? It clearly holds for univariate ...
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### Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?

I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and $$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$ Let ...
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### 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 ...
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### Generating Function Example from class

Example: Consider the sequence $(h_n)$ where $h_n$ is the number of nonnegative integer solutions to $$a_1+a_2+a_3+a_4+a_5=n.,$$ where $a_1$ is even, $a_2$ is odd, $a_3$ is a multiple of $5$, $a_4$ is ...