I understand the whole concept of Rencontres numbers but I can’t understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don’t understand how exactly did the author calculate this recursion […]

Find the number of arrangements of $k \mbox{ }1’$s, $k \mbox{ }2’$s, $\cdots, k \mbox{ }n’$s – total $kn$ cards – so that no same numbers appear consecutively. For $k=2$ we can compute it by using the PIE, and it is $$\frac{1}{2^n} \sum_{i=0}^n (-1)^i \binom{n}{i} (2n-i)! 2^i$$

Number Composition studies the number ways of compositing a number. I wanna know the number of compositions of $m$ with $n$ parts with the size of the max part equal to or less than $k$. Is there a closed form for this problem?

If I have a $w \times h$ matrix where each value is an integer $1 \lt n \lt 20$, how can I count the number of distinct configurations, where $2$ configurations are “distinct” if there is no way to reshuffle the rows and columns that would produce the same matrix? for example these are equal […]

Let $n$ be a number of people. At least two of them may be born on the same day of the year with probability: $$1-\prod_{i=0}^{n-1} \frac{365-i}{365}$$ But what is the probability that at least two of them are born on two consecutive days of the year (considering December 31st and January 1st also consecutive)? It […]

I’m new to graph theory and I don’t plan to become a serious student of graph theory either. My book suggests that there are $11$ non-isomorphic graphs of order $4$, but I can’t see why. I know that the most naive approach is that I can use brute force and draw four vertices and then […]

I’ve already done a few problems such as this, other problems where I’m supposed to find the number of combinations or permutations, subject to certain restrictions. Here’s been my basic strategy: Find $A$ = the number of total solutions (combinations) were there no restrictions. Find $B$ = the number of illegal solutions (solutions that violate […]

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. $$\underbrace{m^{m^{m^{.^{.^{.^{m}}}}}}}_{n-times}$$ Note that one can find a combinatorial description of each one of operators sum, multiplication and exponentiation as follows: $m+n$ is the size of […]

I can’t figure out an algebraic proof for the following identity: $$\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$$ Combinatorical solution: We can see that as choosing some from $n$ and the rest of $k$ from $m$, thus $k$ in total. Or we could just choose $k$ from the union.

I would like to know a way to solve the problem. How can we divide n identical marbles into k distinct with each pile having at most w marbles. I have seen a solution due to Brian Scott on the problem using inclusion-exclusion but I would like one via generating functions. In particular how can […]

Intereting Posts

We all use mathematical induction to prove results, but is there a proof of mathematical induction itself?
Interesting properties of Fibonacci-like sequences?
Finite Sum of Power?
Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$
Fascinating Lampshade Geometry
When is the closure of an open ball equal to the closed ball?
Show uncountable set of real numbers has a point of accumulation
$n$ and $n^5$ have the same units digit?
Applications of the formula expressing roots of a general cubic polynomial
$f'' + f =0$: finding $f$ using power series
Monic and epic implies isomorphism in an abelian category?
toplogical entropy of general tent map
Determing the number of possible March Madness brackets
Proving that Tensor Product is Associative
Limit of Lebesgue measure of interesection of a set $E$ with its translation