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Prove summation related to cycles: For $n \geq r $ there is:$ \sum_{k=1}^{n} {b_r(n,k)x^k=(r-1)!\frac{x^\overline{n}}{(x+1)^\overline{r-1}}} $
Let $b_r(n,k)$ be the number of n-permutations with $k$ cycles, in which numbers $1,2,\dots,r$ are in one cycle.
Prove that for $n \geq r $ there is:
$$
\sum_{k=1}^{n} {b_r(n,k)x^k=(r-1)!\frac{x^\...
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Summing discrete 3d coordinate lengths
Consider a three dimensional discrete coordinate system $(x,y,z)$, where $x,y,z\in$ natural numbers.
The number of digits describing an integer coordinate for each dimension is $l_c=\lfloor log(c) \...
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