# Questions tagged [combinatorial-number-theory]

This is a tag used for number-theoretical questions with combinatorial flavor.

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### Enumeration of $k$-sparse 0/1-vectors of length $N$ [duplicate]

Let $\mathbf{x}$ be a $k$-sparse vector of length $N$ containing $k$ ones. There are $N\choose k$ such vectors and one would need $\log_2 {N\choose k}$ bits to enumerate all of them. Is there an ...
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### Prove or disprove: If [$q\in\mathbb{Q}$ and $a+b = x+y+z = q$] then [$a,b,x,y,z ∈$ either $\mathbb{Q}⊻¬\mathbb{Q}$].

A small observation is that the unit integer $1$ can be split arbitrarily into two pieces: both pieces must exclusively be either rational or irrational (but not a heterogeneous mixture). This ...
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### What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

The classical Euler's gamma function $\Gamma(z)$ can be defined by \Gamma(z)=\lim_{n\to\infty}\frac{n!n^z}{\prod_{k=0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}. \end{...
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### Is this a strict Lower bound on the amount of numbers less than x but coprime to y?

Let $\Lambda(x,y)$ be the relative totient function that counts the amount of numbers less than $x$, which are coprime to $y$. After interpreting Euler's totient function, $\phi(y)$, as a result of ...
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