# 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.

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### Number of vector of length $n$ over ${A,C,T,G}$ that do not have $k$ consecutive $A$'s

In this problem, $k$ is a constant, and we need to find a recursive function that depends only on $n$, i,e $f(n)$. Here is my (wrong) solution: Say there's a valid vector of length $n$. Let's separate ...
• 391
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### Does there exist an oriented graph with fixed amount of vertices and fixed possible indegree and outdegree?

I am considering an oriented graph without loops and multiple edges. The question is: Does there exist an oriented graph with $100$ vertices, where the indegrees of vertices is either $2$ or $10$ and ...
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1 vote
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### Combinatorial Structure and Combinatorial Configuration

I often encounter the terms Combinatorial Structure and Combinatorial Configuration in combinatorial literature. I find few definitions for them, and even when provided, they differ. I am unsure about ...
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### How many 10 letter words are there using the letters a,b,c,d,e,f if [closed]

(a) the letters in the word appear in alphabetical order? (b) each letter occurs at least once and the letters in the word appear in alphabetical order?
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### How many unique circuits can be made given n equal resistors?

Really stumped by this one (I'm not an EE!). Suppose we are given $1, 2, 3, 4, 5$, and $6$ one-ohm resistors. They can be arranged in series, parallel, bridge, and so forth circuits. No dangling ...
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### Compute the value of a double sum

I need some help computing a(n apparently nasty) double sum: $$f(l):=\sum_{j = \frac{l}{2}+1}^{l+1}\sum_{i = \frac{l}{2}+1}^{l+1} \binom{l+1}{j}\binom{l+1}{i} (j-i)^2$$ where $l$ is even. I'm not ...
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### What is the number of facets of a $d$-dimensional cyclic polytope?

A face of a convex polytope $P$ is defined as $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
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### Confused about a counting problem

This question is reproduced from a text by Sheldon Ross: Example 5k. A football team consists of $20$ offensive and $20$ defensive players. The players are to be paired in groups of $2$ for the ...
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### How to "arrange things in groups"? [closed]

I was studying combinatorics in my textbook and I was trying to get through arrangements in groups and here's what I couldn't get Can anyone please help to explain this stuff ?
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### Does there exist at least two sets whose union gives the universe in a certain intersection-closed family of sets?

This is a slightly simplified version of a mathoverflow question without answers. Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is ...
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1 vote
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### Subset of index that minimizes a sum of real values

Given a series of real numbers $c_1, \ldots, c_n$ with $n \in \mathbb{N}$, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
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