1)$ ($m=7$), we show some examples of the previous result for different values of $S$.

1.  $S=50$. Then $\sum_{j=1}^7a_j+a_{0}>S\geq
    \sum_{j=2}^7a_j+2a_{1}$. $k=1$, $S'=S-\sum_{j=1}^7a_j=17$, $q=17$, $r=0$. It results ${\mathbf g}=(17, 8, 6, 5, 5, 5, 3, 1)$.

2.  $S=34$. Then $\sum_{j=3}^7a_j+3a_{2}>S\geq
    \sum_{j=4}^7a_j+4a_{3}$. $k=3$, $S'=S-\sum_{j=3}^7a_j=15$, $q=5$, $r=0$. It results ${\mathbf g}=(5, 5, 5, 5, 5, 5, 3, 1)$.

3.  $S=5$. Then $\sum_{j=8}^7a_j+8a_{7}>S\geq
    \sum_{j=9}^7a_j+9a_{8}$. $k=8$, $S'=S-\sum_{j=8}^7a_j=S=5$, $q=0$, $r=5$. It results ${\mathbf g}=(1, 1, 1, 1, 1, 0, 0, 0)$.

\[propbat2\] Let $E\geq 0$ be a nonnegative integer and let ${\mathbf a}=(a_1, \dots,  a_m)$ be a partition of nonnegative integers. Then there exists a partition of nonnegative integers ${\mathbf e}=(e_1, \dots,  e_{m-1})$ such that $\sum_{i=1}^{m-1} e_i=E$ and ${\mathbf a}\prec' {\mathbf e}$ if and only if $$\label{eqbat2}
 E=\sum_{i=2}^{m}a_i \mbox{ or } E\geq a_1+\sum_{i=3}^{m}a_i.$$

[*Proof.*]{} Assume that there exists ${\mathbf e}=(e_1, \dots,  e_{m-1})$ such that $\sum_{i=1}^{m-1} e_i=E$ and ${\mathbf a}\prec' {\mathbf e}$. Then $e_i\geq a_{i+1},$ $1\leq i \leq m-1$, hence $E\geq \sum_{i=2}^{m}a_i.$

Let us suppose that $E> \sum_{i=2}^{m}a_i.$ Then $\sum_{i=1}^{m-1}(e_i-a_{i+1})>0$, hence there exists $k \in\{1, \dots , m-1\}$ such that $e_k>a_{k+1}$. Thus $k<h=\min\{i \; : e_i<a_i\}$, which means $e_1\geq a_1$ and therefore $$E=e_1+\sum_{i=2}^{m-1}e_i\geq a_1+\sum_{i=3}^{m}a_i.$$

Conversely, let us assume that (\[eqbat2\]) holds. We define $e_i=a_{i+1}$, $2\leq i \leq 