All Questions
7 questions
10
votes
4
answers
1k
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Binomial coefficient in Andrews' partition book
First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and ...
- 4,713
2
votes
1
answer
193
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Bell polynomial with variables 1 and 0
Let $B_{n,k}(x_1,\cdots,x_{n-k+1})$ be the Bell polynomial.
If $x_1=\cdots=x_{n-k+1}=1$, we know that $B_{n,k}(x_1,\cdots,x_{n-k+1})=S(n,k)$, where $S(n,k)$ is the Stirling number of second kind.
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- 34.3k
3
votes
0
answers
144
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Flat polynomials with factors of big height
Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
- 13.4k
13
votes
2
answers
2k
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Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
7
votes
0
answers
264
views
Is every integer $n>1$ the sum of two squares and two central binomial coefficients?
Those integers $\binom{2n}n\ (n=0,1,2,\ldots)$ are called central binomial coefficients. By Stirling's formula,
$$\binom{2n}n\sim \frac{4^n}{\sqrt{n\pi}}\ \ \ \ (n\to+\infty).$$
Of course, the ...
- 15.6k
1
vote
1
answer
363
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limit and combinatorics
Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When $...
- 126
0
votes
0
answers
1k
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Calculating the Shapley value in a weighted voting game.
Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
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