Cumulative abs sums Let $s$ be an $n$-tuple of reals numbers satisfying $\sum_{k=1}^n s_k$. Denote by $\{c_i\}_{i=1}^n$ the permutations of this tuple $s$.
Consider the tuple $$a_{i,j} = \left\{ \begin{array}{l@{,\quad}l}
                                            |{c_{1,j}}|, &   \\
                                            \max\{a_{i,j-1}-c_{i,j,},0\}, &  j=2,\ldots,n
                \end{array}\right..$$
I'd like to how if there is any property from group theory than can be applied to $a_i$.

For $k=1,\ldots,n$ let $p_k=\sum_{i=1}^ks_i$, and fix $k$ such that $p_k$ is minimal; then the true partial sums of $c_{k+1}$ are all non-negative, and hence $a_{k+1,n}=0$. (If $k=n$, $a_{1,n}=0$.)
This is perhaps easiest to see by graphing the partial sums: you’re finding the deepest valley of the original graph and, by shifting the starting point, raising everything by the depth of that valley.
