# Tagged Questions

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} = ... 2answers 330 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 ... 3answers 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 ... 4answers 450 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 ... 3answers 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 ... 0answers 207 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$P(t)$... 4answers 765 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 ... 2answers 468 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 ... 3answers 970 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? ... 0answers 505 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$M$: ... 1answer 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\$, ...
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 ...