# Tagged Questions

**0**

votes

**0**answers

85 views

### Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers
$$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$
For example $f_{n,0} = ...

**10**

votes

**2**answers

325 views

### Inequalities for averaging over partially ordered sets

Let's start from a classical inequality:
If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then
$(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$.
It can be written also in ...

**7**

votes

**3**answers

281 views

### An inequality related to the number of binary strings with no fixed substring

This is basically a repost of this math.se question. At the time I was writing this I thought it has to have a straightforward solution so I posted it there. Now I am not so sure about it being so ...

**2**

votes

**4**answers

422 views

### Sum over integer compositions

Sorry if the question is trivial - are there closed form expressions or good approximations for the sum of a symmetric function taken over all integer compositions (into given number of parts) of a ...

**33**

votes

**3**answers

3k views

### the following inequality is true，but I can't prove it

The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...

**1**

vote

**0**answers

202 views

### Bounding a sum of binomial coefficients in terms of 'the next one'

I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense.
Given a real number $t \geq 2$, call $P(t)$ ...

**15**

votes

**4**answers

760 views

### Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem:
Show, for all integers $1 \leq i \leq k$, that the univariate ...

**10**

votes

**2**answers

464 views

### A property of unimodal sequences

It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies ...

**4**

votes

**3**answers

959 views

### Factorial-inequalities

Let $n>15$ be an integer. Suppose also $n=\sum_{i=1}^n ic_i$, where $c_i$ are non-negative integers. Assume further that $c_1<4$.
Is the following inequality true?
...

**6**

votes

**0**answers

497 views

### Counting permutation matrices in 0,1,2 matrices

Let $M$ be a matrix with entries equal to 0, 1 or 2, such that all of the row and column sums are equal to $c$.
The Van der Waerden bound gives (roughly) the following bound on the permanent of $M$:
...

**10**

votes

**1**answer

1k views

### Sum of difference moduli vs. sum of modulus differences

This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...

**20**

votes

**2**answers

5k views

### Is the Jaccard distance a distance?

Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...