All Questions
13 questions
13
votes
3
answers
1k
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 ...
- 619
11
votes
0
answers
388
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
6
votes
1
answer
372
views
Maximizing a sum minus its maximal summand
This is a followup to a question that appeared on m.SE:
Maximize $\displaystyle f(\pi)=\left(\sum_{i=1}^{n}{i\pi_i}\right)-\max_{1\le i\le n}{(i\pi_i)}$ over permutations $\pi\in S_n$.
The problem ...
- 1,190
3
votes
2
answers
459
views
Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
3
votes
2
answers
255
views
Inequality for Gaussian polynomials III
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
3
votes
1
answer
229
views
Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
- 2,720
3
votes
0
answers
204
views
A connection between the Bell numbers and Bell polynomial
Let $B(n,x) = \sum_{k=0}^n {n\brace k}x^k$ be the Bell polynomials and $B_n = B(n,1)$ be the Bell numbers.
I recently proved a nice relation between the two:
$$
B(n,x)^{1/n}/x \ge B_{n/x}^{x/n},
$$
...
2
votes
1
answer
192
views
Every element of $A$ and $B$ differ in at least $k$ positions
Let $m,n$ be positive integers, $m,n>1$ and $X = \{(x_1,x_2, ..., x_m) \in \mathbb{Z}^m :1 \le x_i \le n, \forall 1 \le i \le m\}$.
$A$ and $B$ are two disjoint subsets of $X$, such that if $a \in ...
- 3,153
2
votes
0
answers
80
views
Inequality on polynomials
Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$.
Given two polynomials $U(q)=\sum_k\alpha_kq^...
- 43.2k
1
vote
2
answers
50
views
Cyclic inequality for 2 dimensional simplex elements
Let $p=(p_{1},p_{2},p_{3})\in\Delta$, with $\Delta:=\lbrace p\in(0,1)^{3}\ |\ p_{1}+p_{2}+p_{3}=1 \rbrace$. I aim to prove (not knowing whether it is true though) that
\begin{equation}
p_{1}^{p_{3}-p_{...
- 289
1
vote
0
answers
105
views
Does this inequality follow from doubly log-concave?
On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$.
One may ...
- 43.2k
0
votes
1
answer
398
views
Forbidden Tripartite Graphs
I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices ...
- 1,129