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Barrett Ruth 2025-08-18 11:39:06 -05:00
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src/content/posts/algorithms/proofs.mdx
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@ -81,7 +81,7 @@ As $S_i<S_{j>i}$, $|S_{j>i}-S_i|=S_{j>i}-S_i$--the sum is unchanged. Rewriting t
$$
\begin{align*}
\sum_{1\leq r\leq n}\sum_{1\leq l<r} X-Y &=\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_r-\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_l
\sum_{1\leq r\leq n}\sum_{1\leq l<r} |X-Y| &=\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_r-\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_l
&=\sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq l<r\leq n} S_l
&=\sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq r\leq n} (n-r)\cdot S_r
\end{align*}