All Questions
Tagged with co.combinatorics inequalities
93 questions
6
votes
0
answers
132
views
Q-analogue of an inequality
Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$.
It is not super-difficult to prove the inequality
$$
\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.
$$
This is actually quite a nice inequality that was ...
- 15.8k
2
votes
0
answers
58
views
Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$
Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
- 6,853
6
votes
1
answer
371
views
An inequality for rearrangement-style sums
The following is a holdover from my math contest days that I never got around
to solve.
We will use the notation $\left[ k\right] $ for the set $\left\{
1,2,\ldots,k\right\} $ whenever $k$ is a ...
- 33.8k
1
vote
1
answer
114
views
additive discrepancy under a multiplicative constraint
Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:
(1). $\sum_{i=1}^K a_i \ge 1/2 + \epsilon$;
(2). $\sum_{i=1}^K d_i \le 1/2 - \...
- 23
0
votes
2
answers
378
views
A symmetric polynomial inequality
I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it....
- 1,776
4
votes
2
answers
431
views
Combination power elementary symmetric polynomial inequality
Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows:
Inequality 1: Let $n>2$ and $1 \le m \le n$...
- 1,776
2
votes
0
answers
125
views
Can this sum be majorized?
Suppose that, we have some real numbers $r_1,r_2,\dots,r_m \in [0,1]$; and we study a sum,
$$
S_1= \sum_{i=1}^m \binom{m}{i}(-1)^{i-1}f(r_i),
$$
for $f:[0,1]\to[0,1]$ a concave bijection. Now, take ...
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
4
votes
2
answers
432
views
How to prove the sum of n squared binomial probabilities does not increase as n increases
Let $F\left( n \right) = \sum\limits_{k = 0}^n {{{\left( {C_n^k{p^k}{{\left( {1 - p} \right)}^{n - k}}} \right)}^2}} $, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.
UPDATE: More general, ...
- 43
9
votes
0
answers
365
views
How to count integer lattice points close to a subspace of $\mathbb R^n$?
Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
- 686
9
votes
1
answer
406
views
Maximize sum of products of binary variable
Let $n>k$ be positive integers, $r>1$ a positive real number, and $A=\{1,2,\dots,n\}$. For $1\leq i\neq j\leq n$, let $a_{i,j}\in\{r,1\}$ be such that $a_{i,j}=r\Leftrightarrow a_{j,i}=1$. ...
- 1,209
1
vote
1
answer
66
views
Inequality about the minimum vertex degree in $k$-uniform hypergraphs
Let $H=(V,E)$ be a $k$-uniform hypergraph with $n$ vertices, that is, $V:=V(H)$ is a $n$-element finite set of vertices and $E:=E(H)\subset\binom{V}{k}$ is a family of $k$-element subsets of $V$.
...
- 139
1
vote
2
answers
111
views
A two-parameter inequality on product of linear terms
I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
$$6\prod_{j=1}^k(...
- 43.2k
0
votes
1
answer
144
views
inequality with binomials by breaking ups
Question. Let $a, b, c\geq0$ be integers. Does this inequality hold?
$$\binom{(a+b+2)(a+c+3)+1}{c+3}\geq\binom{a+c+3}{c+3}(a+b+3)^{c+3}.$$
This inequality happens to appear in some intermediate ...
- 43.2k
3
votes
1
answer
222
views
Intuition for inequality involving permutation and Hamming Cube
Let $C^n=\{0,1\}^n$ be a metric space (Hamming Cube). The distance on $C^n$ is defined by
$$
d(\varepsilon,\varepsilon'):=|\{j:\varepsilon_j\ne\varepsilon'_j\}|,
$$
$\varepsilon=(\varepsilon_1,\...
- 1,245
2
votes
0
answers
204
views
Non-uniform Ray-Chaudhuri-Wilson (generalized Fisher's inequality)
A $t$-design on $v$ points with block size and index $\lambda$ is a collection $\mathcal{B}$ of subsets of a set $V$ with $v$ elements satisfying the following properties:
(a) every $B\in\mathcal{B}$ ...
- 20.2k
1
vote
1
answer
121
views
Probability for high mutual coherence on all subsets of a Gaussian vector set
We examine as set of independent normal vectors:
$$ \forall i \in [N]\triangleq \{1,\dots,N\}:\,\mathbf{x}_{i}\sim\mathcal{N}\left(0,\mathbf{I}_{d}\right)$$
For any $\epsilon>0$ and $K\leq N$, we ...
- 968
4
votes
1
answer
207
views
Upper bound on the number of binary matrices with small rank
I'm looking for the tightest upper bound on the number of different binary matrices $A \in {\{-1,1\}^{m \times n}}$ for which $\mathrm{rank}(A)\leq r$. I'm interested in the regime $1 \ll r \ll m \...
- 968
11
votes
0
answers
291
views
$L_2$ minimizing makespan vs. $L_\infty$ minimizing makespan
There are $n$ positive real numbers. We partition these numbers into $m$ parts, the size of each part is the sum the numbers in this part. Maximum size of the parts is called a makespan of a partition....
- 399
13
votes
4
answers
535
views
Alignment of random points
Whenever I draw randomly about ten points, I see that there will be always 3 points that are "almost" collinear. This observation leads me to considering the following questions:
Question 1: Suppose $...
- 131
20
votes
3
answers
1k
views
mixing convex and concave for convexity
Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$?
$$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
- 43.2k
28
votes
3
answers
1k
views
Inequality for hook numbers in Young diagrams
Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
- 17k
4
votes
1
answer
1k
views
upper bound on derivatives of a function defined on an arc
This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...
- 398
1
vote
0
answers
86
views
Lower bound the ratio of the combinatorial quantities
Suppose $p < q$, $s = p^{d}$ for some fixed $d \in (0,1)$, let $p$ goes to infinity, define the following quantity,
\begin{aligned}
\quad f(j) = \sum_{i = 0}^{\min(j,s)}{s \choose i}{p-s \choose i}{...
- 515
0
votes
1
answer
196
views
For a set of positive integers, is this inequality always true?
The input consists of a set of positive integers $\{b_1,...,b_2\}$ such that $$\sum_{i=1}^nb_i=CK,$$ with $C$ and $K$ two positive integers.
The question is the following, is there $k\in\{1,...,n\}$ ...
- 123
0
votes
1
answer
397
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
5
votes
0
answers
295
views
inequality in a shape of inclusion exclusion formula
I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers $a_1,a_2,...
- 341
14
votes
2
answers
963
views
Conjecture on maximum of symmetric combinatoric function
A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).
(The question was first asked at math.SE, ...
- 155
23
votes
3
answers
3k
views
Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)
Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...
- 986
0
votes
0
answers
131
views
Estimating when does a certain binomial sum exceed an upper bound
Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers
$$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} = 1,f_{...
- 3,463
10
votes
2
answers
496
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 ...
- 2,232
8
votes
3
answers
456
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 ...
- 3,463
6
votes
4
answers
2k
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 ...
- 711
36
votes
3
answers
4k
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 ...
- 363
1
vote
0
answers
399
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)$ ...
- 731
18
votes
4
answers
1k
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 ...
- 6,694
0
votes
0
answers
103
views
An upperbound related to inductively reducing a set by adding the two least elements
I hope this is not too trivial to be asked here, but here it goes anyway. This is just out of curiosity (regarding a problem with graphs but I have "reduced" it to the problem below:)
Let $S_0={\{a_1,...
- 443
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
4
votes
3
answers
2k
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?
$$\frac{n!}{\prod_{i=1}^{n}i^{...
- 3,738
6
votes
0
answers
630
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 $M$:
...
- 1,633
10
votes
1
answer
2k
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 $k$, ...
- 33.8k
62
votes
7
answers
26k
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 ...
- 1,355
21
votes
1
answer
2k
views
Is there a combinatorial proof of Cauchy-Schwarz?
I've only played with this a little for the past day or so, and haven't thought about it too hard, so it might be obvious. Obviously it's not fair to ask for a "combinatorial proof" of an inequality ...
- 12.6k