Mathematics Stack Exchange
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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.

53034 questions
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Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into as many pieces as students come for lunch,…
Stenzel
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In combinatorics, how can one verify that one has counted correctly?

This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly. It's easiest to explain with an example, so I'll give…
Stephen
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7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
nonuser
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How to find a total order with constrained comparisons

There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task? As a partial answer, I know that is possible to…
Jack D'Aurizio
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Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any positive irrational numbers $a\ne e$ such that…
july
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Probability that random moves in the game 2048 will win

I have recently played the game 2048, created by Gabriele Cirulli, which is fun. I suggest trying if you have not. But my brother posed this question to me about the game: If he were to write a script that made random moves in the game 2048, what…
N. Owad
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Why are asymptotically one half of the integer compositions gap-free?

Question summary The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks to understand why. The details A composition of an…
Daniel R
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Number of onto functions

What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements. I found that if $m = 4$ and $n = 2$ the number of onto functions is $14$. But is there a way to generalise this using a…
IcyFlame
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Quadratic reciprocity via generalized Fibonacci numbers?

This is a pet idea of mine which I thought I'd share. Fix a prime $q$ congruent to $1 \bmod 4$ and define a sequence $F_n$ by $F_0 = 0, F_1 = 1$, and $\displaystyle F_{n+2} = F_{n+1} + \frac{q-1}{4} F_n.$ Then $F_n = \frac{\alpha^n -…
Qiaochu Yuan
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How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested in the motivation: Define a signature as a…
MGA
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An illusionist and their assistant are about to perform the following magic trick

Let $k$ be a positive integer. A spectator is given $n=k!+k−1$ balls numbered $1,2,\dotsc,n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as they see fit. The assistant studies the sequence, chooses some block of $k$…
nonuser
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Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
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Exam with $12$ yes/no questions (half yes, half no) and $8$ correct needed to pass, is it better to answer randomly or answer exactly 6 times yes?

In an exam with $12$ yes/no questions with $8$ correct needed to pass, is it better to answer randomly or answer exactly $6$ times yes and 6 times no, given that the answer 'yes' is correct for exactly $6$ questions? I have calculated the…
B.Swan
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How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?

Suppose we have a power tower consisting of $2$ occurring $n$ times: $$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$ How many values can we generate by placing any number of parenthesis? It is fairly simple for the first few values of $n$: There is $1$ value…
barak manos
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Crazy pattern in the simple continued fraction for $\sum_{k=1}^\infty \frac{1}{(2^k)!}$

The continued fraction of this series exhibits a truly crazy pattern and I found no reference for it so far. We have: $$\sum_{k=1}^\infty \frac{1}{(2^k)!}=0.5416914682540160487415778421$$ But the continued fraction is just beautiful: [1, 1, 5, 2,…
Yuriy S
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