Articles of combinatorics

How prove this $2^{2012}-\binom{2012}{1006}<m<2^{2012}$

let $A=\{1,2,3,\cdots,2013\}$,and $A_{1},A_{2},A_{3},\cdots,A_{m}\subset A,A_{i}\neq A_{j},1\le i\le m,1\le j\le n$ and such that:For any $i,j\in \{1,2,3,\cdots,m\},i\neq j$,then $$|A_{i}\bigcap A_{j}|\neq 1$$ prove that $$2^{2012}-\binom{2012}{1006}<m_{\max}<2^{2012}$$ I think this is interesting problem, and This is my frend ask me, I think This problem may be use

What's the shape of this “addsTo” function …?

Note that in this combinatronics question, How many lists of 100 numbers (1 to 10 only) add to 700? I was asking: For an array of 100 numbers, each 1 to 10 inclusive, and the total is T – how many such arrays exist? In fact due to the ASTOUNDING answers there, we now know […]

Graph with girth 5 and exactly $k^2+1$ vertices

I’m doing a homework and I’ve already done the proof that a graph with girth 5 and degree of atleast $k$ has at least $k^2+1$ vertices. Also I’ve been able to found a graph for $k=2$ and $k=3$ but searching over the web I’ve found it’s impossible to found one with $k=4$. How can I […]

Divisors of $2^kp^r$

Let $p$ be an odd prime number. What is the necessary and sufficient condition (in terms of $p$ and $k,r$) such that we can partition the divisors of $2^kp^r$ into two set with equal sum. You may want to LOOK HERE where the special case $2^kp$ is dealed.

Counting the number of partitions having blocks of cardinality 2 and non-distinct elements

Say I have a set of integers $\{1,2,\cdots,n\}$, then there exists $B_n$ partitions of this set where $B_n$ is a Bell number. For instance, there are $B_3$=5 partitions of the set $\{1,2,3\}$: $$ \begin{align} \Pi[\{1,2,3\}] &= \left[\{\{1\},\{2\},\{3\}\}+\{\{1,2\},\{3\}\}\right.\\ &\qquad\left. +\{\{1\},\{2,3\}\}+\{\{1,3\},\{2\}\}+\{\{1,2,3\}\}\right]. \end{align} $$ For a set having an even number of elements, what is the number of […]

Frog on a 3 by 3 matrix

Imagine a frog (f) on a 3 by 3 matrix, with the frog being able to jump to neighboring places from where it sits. So for example if the frog is at $s_{0,1}$, it can subsequently jump to $s_{0,0}, s_{0,2},s_{1,0},s_{1,1}$ and $s_{1,2}$. $$ \left( \begin{array}{ccc} s_{0,0} & f & s_{0,2} \\ s_{1,0} & s_{1,1} & […]

what is the usage of combination $C(r,k)$ where $r$ extends to real number?

Combination is defined as $C(n,k) = \dfrac{n!}{k!(n-k)!}$, where $n$ and $k$ are non-negative integers. Now, the definition can be extended to $C(r,k)$, where $r$ is real number and $k$ is an integer: $$ C(r, k) = \cases{ \frac{r(r-1)\cdots(r-k+1)}{k!} & $k \ge 0$ \\ 1 & $k = 0$ \\ 0 & $k < 0 $} […]

Is there only one counter example in $K_5$ for $R(3,3)$?

Title says it all. And the one that I know is below. (image from wikipedia) My question is: Is there only one counter example in $K_5$ for $R(3,3)$ where $K_5$ is a complete graph of 5 points and $R(s,t)$ is Ramsey number.

Problem with counting method – full house

I’ve never been particularly good at counting, and I was pleasantly surprised to learn today that many typical probability-counting problems can be solved using the equation: $$P(A\cap B) = P(A)P(B|A)$$ and it’s generalisations, rather than by more traditional counting methods. However, my first attempt to apply this equation to a more difficult counting problem has […]

The primes $s$ of the form $6m+1$ and the greatest common divisor of $2s(s-1)$

I had trouble coming up with a good title. Let $\psi(s) = 2s(s-1)$. I write $(\psi(s),\psi(s+2))$ to be the greatest common divisor of $\psi(s)$ and $\psi(s+2)$. Then if $s$ is prime and $(\psi(s),\psi(s+2))=12$ we have that $s=6m+1$ for some positive integer $m$. Conversely if $s=6m+1$ is prime then $(\psi(s),\psi(s+2))=12$. I would like to solve this […]