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 ${\mathbf g}=(g_1, \dots,  g_{m+1})$ such that $\sum_{i=1}^{m+1} g_i=S$ and ${\mathbf g}\prec'{\mathbf a}$.

[*Proof.*]{} Put $a_0=+\infty$, $a_{m+1}=-\infty$. Then, $$\sum_{j=i}^ma_j+ia_{i-1}\geq \sum_{j=i+1}^ma_j+(i+1)a_{i}, \quad 1\leq i \leq m+1,$$ and $
S\geq (m+2)a_{m+1}
$. Let $
k=\min\{i \in \{1, \dots, m+1\}\; : \; S\geq \sum_{j=i+1}^ma_j+(i+1)a_{i}\}
$, i.e., $$\sum_{j=k}^ma_j+ka_{k-1}>S\geq \sum_{j=k+1}^ma_j+(k+1)a_{k}=\sum_{j=k}^ma_j+ka_{k}.$$ Let $S'=S-\sum_{j=k}^ma_j$. Then $ka_{k-1}> S'\geq ka_{k}$. Let $q$ and $r$ be the quotient and the remainder of the euclidian division of $S'$ by $k$, i.e., $S'=kq+r$ with $0\leq r <k$. Then $a_{k-1}> q\geq a_{k}$. Observe that if $k\leq m$, then $a_k\geq 0$, and if $k=m+1$, then $S'=S\geq 0$, hence $q\geq 0$.

Let us define $$\begin{array}{ll}
g_i=q+1, & 1\leq i \leq r,\\
g_i=q, & r+1\leq i \leq k,\\
g_i=a_{i-1}, & k+1\leq i \leq m+1. \\
\end{array}$$ Then $g_1\geq\dots\geq g_{m+1}\geq 0$, $\sum_{i=1}^{m+1} g_i=S$ and $$\label{ga}
\begin{array}{ll}
g_{i}\leq q+1\leq a_{k-1}\leq a_i, & 1\leq i \leq k-1,\\
g_{i+1}=a_{i}, & k\leq i \leq m.\\
\end{array}$$ Let $h=\min\{i\; : \; a_i<g_i\}$. From (\[ga\]) we derive that $h\geq k$ and $a_{i}=g_{i+1}$, $h\leq i \leq m$. Therefore, $(g_1,\dots,g_{m+1})\prec'{\mathbf a}$.

$\Box$

Given the partition ${\mathbf a}=(8, 6, 5, 5, 5, 3, 