Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially 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.

Combinatorics is the study of finite or countable discrete structures — especially enumerative combinatorics: how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. This tag can be used for questions about permutations, combinations, partially ordered sets, bijection proofs, and generating functions.

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To learn more about enumerative combinatorics, see Wikipedia or Richard P. Stanley's Enumerative Combinatorics.

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How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}\right) \div .4$. If we remove the restriction…
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Do men or women have more brothers?

Do men or women have more brothers? I think women have more as no man can be his own brother. But how one can prove it rigorously? I am going to suggest some reasonable background assumptions: There are a large number of individuals, of whom half…
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Why can a Venn diagram for $4+$ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could…
Larry Wang
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Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to board a plane that seats 100. The first person in…
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There are apparently $3072$ ways to draw this flower. But why?

This picture was in my friend's math book: Below the picture it says: There are $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen. I know it's based on combinatorics, but I don't know how to…
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Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in arbitrary order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every person has sat either to the right or to the left…
Vincent Tjeng
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Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which this property holds? If this question is too broad,…
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Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a…
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Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian Art Problem. For…
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Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on

Popular mathematics folklore provides some simple tools enabling us compactly to describe some truly enormous numbers. For example, the number $10^{100}$ is commonly known as a googol, and a googol plex is $10^{10^{100}}$. For any number $x$, we…
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Cutting sticks puzzle

This was asked on sci.math ages ago, and never got a satisfactory answer. Given a number of sticks of integral length $ \ge n$ whose lengths add to $n(n+1)/2$. Can these always be broken (by cuts) into sticks of lengths $1,2,3, \ldots…
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Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing terms gives us $$\sum_{k=0}^n…
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Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I've now also asked this question on mathoverflow. Notes: Obviously I'm ignoring the triangle of side $0$. Because the area of the triangle has to be a…
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Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the real exam. After each mock exam the teacher tells…
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$6!\cdot 7!=10!$. Is there a natural bijection between $S_6\times S_7$ and $S_{10}$?

Aside from $1!\cdot n!=n!$ and $(n!-1)!\cdot n! = (n!)!$, the only nontrivial product of factorials known is $6!\cdot 7!=10!$. One might naturally associate these numbers with the permutations on $6, 7,$ and $10$ objects, respectively, and hope that…
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