## New answers tagged partitions

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### "Geodesic coherent" partition of a graph

Pilipczuk and Siebertz proved that every planar graph has such a partition with an even stronger property. Namely, each part $V_i$ is a geodesic path, and the graph obtained by contracting each part ...

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### Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$

Notice that for $i\in\{0,1,\dots,2^{n-1}+n\}$ we have
$$a(i+1,2^{n-1}+n) = 2^{2^{n-1}+n+1} - 1 - 2^{2^{n-1}+n-i}.$$
Then the sum in question can be easily computed:
\begin{split}
& a(1,2^{n-1}+n)+\...

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