For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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18
votes
3answers
567 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...
18
votes
1answer
409 views

Special about 2015 with conjecture?

In the question Is there something special about 2015? Jack D'Aurizio found a nice result: $$\dfrac{1^2+2^2+\cdots +77^2}{77}=2015.$$ Then I was wondering: Is there a rectangular table $(7\times ...
18
votes
1answer
743 views

How to prove a double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
18
votes
3answers
602 views

Counting binary sequences with no more than $2$ equal consecutive numbers

I invented the following problem, but I can't solve it. There are $n$ $1$'s and $n$ $0$'s and my question is what is the number of ways to arrange them avoiding $3$ equal consecutive numbers. So for ...
18
votes
2answers
518 views

Maximizing curious symmetric function from simple combinatorics

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). (This question has been posted at ...
18
votes
1answer
321 views

Count the number of bases in a subset

Consider $\mathbb{R}^n$ as a vector space over $\mathbb{R}$. Consider the subset $\mathrm{S}^n = \{(x_1,\ldots,x_n) \in \mathbb{R}^n | x_i = 0 \; \mathrm{or} \; 1\;\forall i = 1,\ldots,n\}$. How many ...
18
votes
2answers
1k views

The Weaver Android app $\rightarrow$ cute combinatorics problem

There's an Android puzzle app called "The Weaver". My question is why every level seems to be solvable in far fewer moves than one might naively think. Here's a link for people who want to play along ...
18
votes
1answer
552 views

Probability that two randomly chosen permutations will generate $S_n$.

Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$. Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle ...
18
votes
1answer
1k views

Number of moves to solve a flood-it/sock-dye game

[ Question based on the sock dye game ] [ Update: It appears that this game is better known as "Flood it" and is NP-hard. Also, "the number of moves required to flood the whole board is $\Omega(n)$ ...
18
votes
1answer
375 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
18
votes
1answer
283 views

Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
17
votes
5answers
2k views

Probability of random integer's digits summing to 12

What is the probability that a random integer between 1 and 9999 will have digits that sum to 12? As a user suggested, I could make a spreadsheet and count them, but is there a quicker way to do ...
17
votes
7answers
3k views

What is combinatorics?

I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is. Could anyone give me a definition/explanation of combinatorics, ...
17
votes
5answers
1k views

Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially

$$\text{Evaluate } \sum_{k=1}^n k^2 \text{ and } \sum_{k=1}^{n}k(k+1) \text{ combinatorially.}$$ For the first one, I was able to express $k^2$ in terms of the binomial coefficients by ...
17
votes
4answers
6k views

Generating functions for combinatorics

I have no formal education in generating functions, but, based on another question, I have seen that they can be powerful for combinatorics. Are there a few general principles for using generating ...
17
votes
1answer
2k views

Why are there only a few known Ramsey numbers?

Can someone explain in a simple way, why there are so few known exact Ramsey Numbers? I guess it's because there are no efficient algorithms for this task, but are there so many combinations to test? ...
17
votes
1answer
891 views

Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s - total $kn$ cards.

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 ...
17
votes
4answers
396 views

Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$

Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$ I know that there are tools in Number theory to proves the required but I want to use the tool that says that if you ...
17
votes
5answers
2k views

Incredible Blackjack Hand

Last Saturday night I played at Bally's in Atlantic City and got a hand I could not believe. Dealer had 9 and I was dealt 2 8s. I split the 8s and was given a third card. It was an 8 so I split them ...
17
votes
3answers
3k views

Gay Speed Dating Problem

Here's an interesting problem that I came up with the other night. With straight speed dating, (assuming the number of men and women are equal) the number of iterations that need to be made before ...
17
votes
4answers
773 views

Determinant of a generalized Pascal matrix

Let $M$ denote the infinite matrix defined recursively by $$ M_{ij} = \begin{cases} 1, & \text{if } i=1 \text{ and } j=1; \\ aM_{i-1,j}+bM_{i,j-1}+cM_{i-1,j-1}, & ...
17
votes
5answers
1k views

Lower bound in algorithmic puzzle

Puzzle: there are $n$ computers most of which are good; the others may be bad ("most" in the strict sense: there are strictly more good computers than bad ones). You may ask any computer $A$ about the ...
17
votes
3answers
7k views

Expanding and understanding the poison pills riddle

You might have heard of the riddle that asks you to identify one fake pill (poisoned) among 12 pills of identical appearance, with the fake pill being either lighter or heavier than the others. You ...
17
votes
4answers
553 views

$N$ perfect logicians wearing hats

I once came across the following riddle: (assume $N$ to be extremely large) There are $N$ perfect logicians arranged in a vertical row. They are allowed to strategize before the game, during the ...
17
votes
3answers
482 views

Prove an inequality on $l^2$ sequences over $F_2$

Denote $F_2$ the free non-abelian group on two letters $a, b$. Note that any element in $F_2$ is just a word formed by letters from the set $\{a,b,a^{-1},b^{-1}\}$, and the group structure is given ...
17
votes
5answers
618 views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ (Dixon's identity)

I found the following formula in a book without any proof: $$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$ I don't know how to prove this at all. Could you show me how ...
17
votes
2answers
311 views

A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses ...
17
votes
1answer
292 views

Is there a combinatoric identity for the multiplicities of the following set?

Are you ready for some psychedelic pictures? Define the multiset$$S_n=\left\{\sum_{j=1}^n(-1)^{\left\lfloor(k-1)/2^{j-1}\right\rfloor}u_n^j\mbox{ for }1\leq k\leq2^n\right\}$$ where ...
17
votes
2answers
343 views

5 moving points in plane, one goes to infinity

Suppose we have $5$ points in plane, each lying on a line for which no three of these lines intersect in one point, and also non of these $5$ points is an intersection point of two lines. At time ...
16
votes
5answers
726 views

Combinatorially showing $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$

I am trying to show that $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$. I found that using stirling's approximation, I can get: $$ \lim_{n\to \infty}{\frac{2n\choose n}{4^n}}= \lim_{n\to ...
16
votes
3answers
1k views

Alternating sum of squares of binomial coefficients

I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n}\choose{0}$$^2$ - ${n}\choose{1}$$^2$ + ${n}\choose{2}$$^2$ + ... + ...
16
votes
4answers
815 views

How to solve $\binom{n}{1}^2+2\binom{n}{2}^2 + 3\binom{n}{3}^2 + 4\binom{n}{4}^2+\cdots + n\binom{n}{n}^2$?

I have tried something to solve the series $$\binom{n}{1}^2+2\binom{n}{2}^2 + 3\binom{n}{3}^2 + 4\binom{n}{4}^2+\cdots + n\binom{n}{n}^2.$$ My approach is : $$(1+x)^n=\binom{n}{0} + \binom{n}{1}x + ...
16
votes
4answers
466 views

Prove that $n! \equiv \sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(n-k+r)^{n} $

Basically I had some fun doing this: 0 1 1 6 7 6 8 12 19 6 27 18 37 6 64 24 61 125 etc. starting with ...
16
votes
6answers
434 views

Generating function for binomial coefficients $\binom{2n+k}{n}$ with fixed $k$

Prove that $$ \frac{1}{\sqrt{1-4t}} \left(\frac{1-\sqrt{1-4t}}{2t}\right)^k = \sum\limits_{n=0}^{\infty}\binom{2n+k}{n}t^n, \quad \forall k\in\mathbb{N}. $$ I tried already by induction over $k$ but i ...
16
votes
3answers
2k views

Why there are $11$ non-isomorphic graphs of order $4$?

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 ...
16
votes
7answers
2k views

Probability that the sum of three integer numbers (from 1 to 100) is more than 100

I have an urn with $100$ balls. Each ball has a number in it, from $1$ to $100$. I take three balls from the urn without putting the balls again in the urn. I sum the three numbers obtained. What's ...
16
votes
3answers
799 views

Reference for combinatorial game theory.

What is a good reference material for elementary combinatorial game theory? By combinatorial game theory I mean chiefly the study of zero-sum, deterministic two-player games (perhaps even more ...
16
votes
3answers
447 views

Is it possible to uniquely number faces of a hexagonal grid with consecutive numbers?

You have a grid of regular hexagons. The aim of the game is to have each hex contain the numbers 1-6 on its edges. Each edge must also be connected to another edge that has a value one higher and ...
16
votes
3answers
2k views

Consecutive birthdays probability

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 ...
16
votes
2answers
1k views

What's the General Expression For Probability of a Failed Gift Exchange Draw

My family does a gift exchange every year at Christmas. There are five couples and we draw names from a hat. If a person draws their own name, or the name of their spouse, all the names go back in a ...
16
votes
4answers
658 views

Decreasing integers on the blackboard

There are $n\geq 2$ copies of an integer $k>0$ written on the blackboard. A move consists of choosing an integer $m>0$ on the blackboard, and replacing it as well as one other integer on the ...
16
votes
4answers
531 views

Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
16
votes
2answers
823 views

Shortest sequence containing all permutations

Given an integer $n$, define $s(n)$ to be the length of the shortest sequence $S = (a_1, \cdots a_{s(n)})$ such that every permutation of $\{1,\cdots,n\}$ is a subsequence of $S$. If $n=1$, then $S = ...
16
votes
1answer
954 views

How many different shapes can I make with this toy?

I have the following toy, perhaps some of you have seen it before. It consists of a bunch of cubes with an elastic string in the middle. You can bend it into different shapes like this: Or this: ...
16
votes
3answers
368 views

Distribution of a maximum

Randomly select $n$ numbers from ${\{1,2,\dots,m\}}$ without replacement, and order the chosen elements increasingly: $X_1 < X_2 < \dots < X_n$ And we can view each $X_i$ as a random ...
16
votes
5answers
24k views

Number of ways to write n as a sum of k nonnegative integers

How many ways can I write a positive integer $n$ as a sum of $k$ nonnegative integers up to commutativity? For example, I can write $4$ as $0+0+4$, $0+1+3$, $0+2+2$, and $1+1+2$. I know how to ...
16
votes
1answer
2k views

Combinatorial proof of $\sum\limits_{k=0}^n {n \choose k}3^k=4^n$

Using the following equation: $$\sum_{k=0}^n {n \choose k}3^k=4^n$$ I need to prove that both sides of the equation solve the same combinatorial problem. It's easy to see that the right side of the ...
16
votes
1answer
545 views

How many strategies are there for this puzzle where one of n logicians must call his own hat's color among n?

$n$ logicians are wearing hats which can be of $n$ different colors. Each logician can see the colors of all hats except his own. The logicians must simultaneously call out a color; they win if at ...
16
votes
1answer
392 views

On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
16
votes
2answers
540 views

Number of factorizations of distinct factors

Let $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$ an integer with $p_i$ prime and $e_i \in \mathbb N$. The prime factorization can assumed to be known, i.e., we already know $p_1, \dotsc, p_k$ and $e_1, ...